School Subjects

Branch of Philosophy

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<b><span style="color: #003bb0; font-family: "Times New Roman","serif"; font-size: 18.0pt; mso-fareast-font-family: "Times New Roman";">Introduction to the
Five Branches of Philosophy<o:p></o:p></span></b></div>
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<span style="font-family: 'Times New Roman', serif; font-size: 13.5pt;">Philosophy
can be divided into five branches which address the following questions:<o:p></o:p></span></div>
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<span style="font-family: "Times New Roman","serif"; font-size: 12.0pt; mso-fareast-font-family: "Times New Roman";"><a href="http://www.importanceofphilosophy.com/Metaphysics_Main.html"><b><span style="color: #003bb0;">Metaphysics</span></b></a><o:p></o:p></span></div>
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<span style="font-family: "Times New Roman","serif"; font-size: 12.0pt; mso-fareast-font-family: "Times New Roman";">Study of Existence<o:p></o:p></span></div>
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<span style="font-family: "Times New Roman","serif"; font-size: 12.0pt; mso-fareast-font-family: "Times New Roman";">What's out there?<o:p></o:p></span></div>
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<span style="font-family: "Times New Roman","serif"; font-size: 12.0pt; mso-fareast-font-family: "Times New Roman";"><a href="http://www.importanceofphilosophy.com/Epistemology_Main.html"><b><span style="color: #003bb0;">Epistemology</span></b></a><o:p></o:p></span></div>
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<span style="font-family: "Times New Roman","serif"; font-size: 12.0pt; mso-fareast-font-family: "Times New Roman";">Study of Knowledge<o:p></o:p></span></div>
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<span style="font-family: "Times New Roman","serif"; font-size: 12.0pt; mso-fareast-font-family: "Times New Roman";">How do I know about it?<o:p></o:p></span></div>
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<span style="font-family: "Times New Roman","serif"; font-size: 12.0pt; mso-fareast-font-family: "Times New Roman";"><a href="http://www.importanceofphilosophy.com/Ethics_Main.html"><b><span style="color: #003bb0;">Ethics</span></b></a><o:p></o:p></span></div>
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<span style="font-family: "Times New Roman","serif"; font-size: 12.0pt; mso-fareast-font-family: "Times New Roman";">Study of Action<o:p></o:p></span></div>
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<span style="font-family: "Times New Roman","serif"; font-size: 12.0pt; mso-fareast-font-family: "Times New Roman";">What should I do?<o:p></o:p></span></div>
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<span style="font-family: "Times New Roman","serif"; font-size: 12.0pt; mso-fareast-font-family: "Times New Roman";"><a href="http://www.importanceofphilosophy.com/Politics_Main.html"><b><span style="color: #003bb0;">Politics</span></b></a><o:p></o:p></span></div>
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<span style="font-family: "Times New Roman","serif"; font-size: 12.0pt; mso-fareast-font-family: "Times New Roman";">Study of Force<o:p></o:p></span></div>
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<span style="font-family: "Times New Roman","serif"; font-size: 12.0pt; mso-fareast-font-family: "Times New Roman";">What actions are permissible?<o:p></o:p></span></div>
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<span style="font-family: "Times New Roman","serif"; font-size: 12.0pt; mso-fareast-font-family: "Times New Roman";"><a href="http://www.importanceofphilosophy.com/Esthetics_Main.html"><b><span style="color: #003bb0;">Esthetics</span></b></a><o:p></o:p></span></div>
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<span style="font-family: "Times New Roman","serif"; font-size: 12.0pt; mso-fareast-font-family: "Times New Roman";">Study of Art<o:p></o:p></span></div>
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<span style="font-family: "Times New Roman","serif"; font-size: 12.0pt; mso-fareast-font-family: "Times New Roman";">What can life be like?<o:p></o:p></span></div>
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<span style="font-family: 'Times New Roman', serif; font-size: 13.5pt;">There
is a hierarchical relationship between these branches as can be seen in the</span><span style="font-family: 'Times New Roman', serif; font-size: 13.5pt;"> </span><span style="font-family: 'Times New Roman', serif; font-size: 13.5pt;"><a href="http://www.importanceofphilosophy.com/Chart.html"><span style="color: #003bb0; mso-bidi-font-size: 11.0pt;">Concept Chart</span></a>. At the root is</span><span style="font-family: 'Times New Roman', serif; font-size: 13.5pt;"> </span><span style="font-family: 'Times New Roman', serif; font-size: 13.5pt;"><a href="http://www.importanceofphilosophy.com/Metaphysics_Main.html"><span style="color: #003bb0; mso-bidi-font-size: 11.0pt;">Metaphysics</span></a>, the
study of existence and the nature of existence. Closely related is</span><span style="font-family: 'Times New Roman', serif; font-size: 13.5pt;"> </span><span style="font-family: 'Times New Roman', serif; font-size: 13.5pt;"><a href="http://www.importanceofphilosophy.com/Epistemology_Main.html"><span style="color: #003bb0; mso-bidi-font-size: 11.0pt;">Epistemology</span></a>, the study
of</span><span style="font-family: 'Times New Roman', serif; font-size: 13.5pt;"> </span><span style="font-family: 'Times New Roman', serif; font-size: 13.5pt;"><a href="http://www.importanceofphilosophy.com/Epistemology_Knowledge.html"><span style="color: #003bb0; mso-bidi-font-size: 11.0pt;">knowledge</span></a></span><span style="font-family: 'Times New Roman', serif; font-size: 13.5pt;"> </span><span style="font-family: 'Times New Roman', serif; font-size: 13.5pt;">and how we know about reality and existence.
Dependent on Epistemology is</span><span style="font-family: 'Times New Roman', serif; font-size: 13.5pt;"> </span><span style="font-family: 'Times New Roman', serif; font-size: 13.5pt;"><a href="http://www.importanceofphilosophy.com/Ethics_Main.html"><span style="color: #003bb0; mso-bidi-font-size: 11.0pt;">Ethics</span></a>, the study of
how man should act. Ethics is dependent on Epistemology because it is
impossible to make choices without knowledge. A subset of Ethics is<a href="http://www.importanceofphilosophy.com/Politics_Main.html"><span style="color: #003bb0; mso-bidi-font-size: 11.0pt;">Politics</span></a>: the study
of how men should interact in</span><span style="font-family: 'Times New Roman', serif; font-size: 13.5pt;"> </span><span style="font-family: 'Times New Roman', serif; font-size: 13.5pt;"><a href="http://www.importanceofphilosophy.com/Politics_ProperGovernment.html"><span style="color: #003bb0; mso-bidi-font-size: 11.0pt;">a proper society</span></a></span><span style="font-family: 'Times New Roman', serif; font-size: 13.5pt;"> </span><span style="font-family: 'Times New Roman', serif; font-size: 13.5pt;">and what constitutes proper.<a href="http://www.importanceofphilosophy.com/Esthetics_Main.html"><span style="color: #003bb0; mso-bidi-font-size: 11.0pt;">Esthetics</span></a>, the study
of</span><span style="font-family: 'Times New Roman', serif; font-size: 13.5pt;"> </span><span style="font-family: 'Times New Roman', serif; font-size: 13.5pt;"><a href="http://www.importanceofphilosophy.com/Esthetics_Art.html"><span style="color: #003bb0; mso-bidi-font-size: 11.0pt;">art</span></a></span><span style="font-family: 'Times New Roman', serif; font-size: 13.5pt;"> </span><span style="font-family: 'Times New Roman', serif; font-size: 13.5pt;">and</span><span style="font-family: 'Times New Roman', serif; font-size: 13.5pt;"> </span><span style="font-family: 'Times New Roman', serif; font-size: 13.5pt;"><a href="http://www.importanceofphilosophy.com/Esthetics_SenseOfLife.html"><span style="color: #003bb0; mso-bidi-font-size: 11.0pt;">sense of life</span></a></span><span style="font-family: 'Times New Roman', serif; font-size: 13.5pt;"> </span><span style="font-family: 'Times New Roman', serif; font-size: 13.5pt;">is slightly separate, but depends on
Metaphysics, Epistemology, and Ethics.<o:p></o:p></span></div>
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Branch of Philosophy

Introduction to the Five Branches of Philosophy Philosophy can be divided into five branches which address the following questions: ...

Locomotor and Nonlocomotor

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<span style="font-family: Verdana, sans-serif; font-size: 10pt;">Developing
physical skills is a prerequisite to being active for life and physical
education class is where it happens!<o:p></o:p></span></div>
<div style="text-align: justify;">
<span style="font-family: Verdana, sans-serif; font-size: 10pt;">Students need to develop basic
physical skills and a degree of competence which can lead to a desire to
continually participate in physical activity. Their perceptions about physical
activity, often formed in physical education, provide the key to future motivation
and participation. In order for students to willingly participate in an
activity, they need to feel successful at that activity. Activities should be
geared to the level of the individual, and should be developmentally
appropriate.<o:p></o:p></span></div>
<div style="text-align: justify;">
<span style="font-family: Verdana, sans-serif; font-size: 10pt;">In schools we assist students
in learning, developing and combining the basic skills of movement so they can
be applied to a variety of activity situations in their lifetime. By combining
and applying basic skills in a variety of activity settings students can learn
and attain a degree of competence. This application of the basic skills is
accomplished through practice and participation. The benefits of physical
activity increase proportionally in relation to the time and intensity
provided. So maximizing the amount of time available for practice and
participation within the school context should be considered. Students will be
more inclined to participate with enthusiasm and vigour if the classroom
atmosphere is one which focuses on creating a desire to increase student participation.
Development of skills will increase if students have more opportunity to
participate in quality learning experiences which focus on motivating, fun,
inclusive activities. They will be more willing to be active for a lifetime.<o:p></o:p></span></div>
<div style="text-align: justify;">
<span style="font-family: Verdana, sans-serif; font-size: 10pt;">General Outcomes in
"A" - Activity is the basis for physical education as this program is
all about moving!<o:p></o:p></span></div>
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<span style="font-family: Verdana, sans-serif; font-size: 10pt;"><br /></span></div>
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<b><span style="font-family: Verdana, sans-serif; font-size: 10pt;">General Outcome A (or
GO"A")</span></b><span class="apple-converted-space"><span style="font-family: Verdana, sans-serif; font-size: 10pt;"> </span></span><span style="font-family: Verdana, sans-serif; font-size: 10pt;">Students
will acquire skills through a variety of developmentally appropriate movement
activities; dance, games, types of gymnastics, individual activities and
activities in an alternative environment; e.g., aquatics and outdoor pursuits.<o:p></o:p></span></div>
<div style="text-align: justify;">
<span style="font-family: Verdana, sans-serif; font-size: 10pt;">General Outcome A looks at two
areas which relate to physical activity. These are listed under two curriculum
organizers: Basic Skills and Application of Basic Skills.<o:p></o:p></span></div>
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<span style="font-family: Verdana, sans-serif; font-size: 10pt;"><br /></span></div>
<div style="text-align: justify;">
<b><span style="font-family: Verdana, sans-serif; font-size: 10pt;">Basic skills:<span class="apple-converted-space"> </span></span></b><span style="font-family: Verdana, sans-serif; font-size: 10pt;">The basic skills of
physical education have been organized into three categories:<o:p></o:p></span></div>
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<span style="font-family: Verdana, sans-serif; font-size: 10pt;"><br /></span></div>
<div style="text-align: justify;">
<b><span style="font-family: Verdana, sans-serif; font-size: 10pt;">Locomotor skills:</span></b><span style="font-family: Verdana, sans-serif; font-size: 10pt;">These are the skills students
develop while travelling. Examples of locomotor skills include; e.g., walking,
running, hopping, jumping, leaping, rolling, skipping, galloping, climbing,
sliding, propulsion through water.<o:p></o:p></span></div>
<div style="text-align: justify;">
<span style="font-family: Verdana, sans-serif; font-size: 10pt;">•
Walk - using the feet to advance by steps.  
<o:p></o:p></span></div>
<div style="text-align: justify;">
<span style="font-family: Verdana, sans-serif; font-size: 10pt;">•
Run - move fast by using the feet, with one foot off the ground at any given
time. <o:p></o:p></span></div>
<div style="text-align: justify;">
<span style="font-family: Verdana, sans-serif; font-size: 10pt;">•
Hop - push off 1 foot, land on same foot. 
Vertical or horizontal. <o:p></o:p></span></div>
<div style="text-align: justify;">
<span style="font-family: Verdana, sans-serif; font-size: 10pt;">•
Jump - take off 1 or 2 feet, land on 2 feet. 
Vertical or horizontal. <o:p></o:p></span></div>
<div style="text-align: justify;">
<span style="font-family: Verdana, sans-serif; font-size: 10pt;">•
Leap - take off 1 foot, cover a large distance, land on other foot.  Use several running steps to lead into the <o:p></o:p></span></div>
<div style="text-align: justify;">
<span style="font-family: Verdana, sans-serif; font-size: 10pt;">leap.
<o:p></o:p></span></div>
<div style="text-align: justify;">
<span style="font-family: Verdana, sans-serif; font-size: 10pt;">•
Gallop - feet face forward, step forward & close with the trail foot, step
& close... (same foot always leads). <o:p></o:p></span></div>
<div style="text-align: justify;">
<span style="font-family: Verdana, sans-serif; font-size: 10pt;">•
Skip - step forward, hop on that foot while bringing other foot forward to step
and hop and switch. <o:p></o:p></span></div>
<div style="text-align: justify;">
<span style="font-family: Verdana, sans-serif; font-size: 10pt;">•
Slide - step to the side, close with other foot, step to the side again, close
with other...<o:p></o:p></span></div>
<div style="text-align: justify;">
<span style="font-family: Verdana, sans-serif; font-size: 10pt;"><br /></span></div>
<div style="text-align: justify;">
<b><span style="font-family: Verdana, sans-serif; font-size: 10pt;">Nonlocomotor skills:</span></b><span class="apple-converted-space"><span style="font-family: Verdana, sans-serif; font-size: 10pt;"> </span></span><span style="font-family: Verdana, sans-serif; font-size: 10pt;">These
are the skills students develop while moving but remaining in one spot. These
may include; e.g., turning, twisting, swinging, balancing, bending, landing,
stretching, curling, hanging.<o:p></o:p></span></div>
<div style="text-align: justify;">
<span style="font-family: Verdana, sans-serif; font-size: 10pt;">•
Stretch - extend the limbs or muscles, or the entire body. <o:p></o:p></span></div>
<div style="text-align: justify;">
<span style="font-family: Verdana, sans-serif; font-size: 10pt;">•
Bend - movement that causes the formation of a curve. <o:p></o:p></span></div>
<div style="text-align: justify;">
<span style="font-family: Verdana, sans-serif; font-size: 10pt;">•
Twist - rotate body or body parts around a stationary base (twist upper body
back and forth, keep feet still). <o:p></o:p></span></div>
<div style="text-align: justify;">
<span style="font-family: Verdana, sans-serif; font-size: 10pt;">•
Turn - partial or full rotation of the body while shifting the base of support
(spin to make yourself dizzy). <o:p></o:p></span></div>
<div style="text-align: justify;">
<span style="font-family: Verdana, sans-serif; font-size: 10pt;">•
Collapse – to fall down. <o:p></o:p></span></div>
<div style="text-align: justify;">
<span style="font-family: Verdana, sans-serif; font-size: 10pt;">•
Punch - use forceful actions with various body parts - hands, feet, knees,
elbows. <o:p></o:p></span></div>
<div style="text-align: justify;">
<span style="font-family: Verdana, sans-serif; font-size: 10pt;">•
Slash - longer movements but powerful like punches (picture a slashing sword). <o:p></o:p></span></div>
<div style="text-align: justify;">
<span style="font-family: Verdana, sans-serif; font-size: 10pt;">•
Push - try to move away by pressure. <o:p></o:p></span></div>
<div style="text-align: justify;">
<span style="font-family: Verdana, sans-serif; font-size: 10pt;">•
Pull - exert force on object to move it towards source of force. <o:p></o:p></span></div>
<div style="text-align: justify;">
<span style="font-family: Verdana, sans-serif; font-size: 10pt;">•
Swing - move or walk in a to and fro or swaying manner. <o:p></o:p></span></div>
<div style="text-align: justify;">
<span style="font-family: Verdana, sans-serif; font-size: 10pt;">•
Sway - swing unsteadlily; rock by moving back and forth sideways.<o:p></o:p></span></div>
<div style="text-align: justify;">
<span style="font-family: Verdana, sans-serif; font-size: 10pt;"><br /></span></div>
<div style="text-align: justify;">
<b><span style="font-family: Verdana, sans-serif; font-size: 10pt;">Manipulative:<span class="apple-converted-space"> </span></span></b><span style="font-family: Verdana, sans-serif; font-size: 10pt;">These
are the skills developed while using an implement. These may include:
receiving; e.g., catching, collecting: retaining; e.g., dribbling, carrying,
bouncing, trapping: sending; e.g., throwing, kicking, striking.<o:p></o:p></span></div>
<div style="text-align: justify;">
<b><span style="font-family: Verdana, sans-serif; font-size: 10pt;">Application of Basic
skills:<span class="apple-converted-space"> </span></span></b><span style="font-family: Verdana, sans-serif; font-size: 10pt;">The
application of the basic skills, as listed above, should be demonstrated
through all five dimensions. Dimensions are the categories of activities
through which students can achieve the outcomes and which teachers use for
planning.<o:p></o:p></span></div>
<div style="text-align: justify;">
<b><span style="font-family: Verdana, sans-serif; font-size: 10pt;">Application of Basic Skills in an Alternative Environment -</span></b><span class="apple-converted-space"><span style="font-family: Verdana, sans-serif; font-size: 10pt;"> </span></span><span style="font-family: Verdana, sans-serif; font-size: 10pt;">Students participate in a variety of land-
or water-based activities. The selection of activities will depend on the
climate, resources and facilities available in the school or community.<o:p></o:p></span></div>
<div style="text-align: justify;">
<b><span style="font-family: Verdana, sans-serif; font-size: 10pt;">Application of Basic Skills in Dance -</span></b><span style="font-family: Verdana, sans-serif; font-size: 10pt;">Students
participate in a wide variety of dance experiences to enhance development of
creative, expressive and rhythmical movements. Students gain awareness of and
respect for their own and other cultures, enhance cooperation skills and
develop skills that are transferable to social situations in the community.<o:p></o:p></span></div>
<div style="text-align: justify;">
<b><span style="font-family: Verdana, sans-serif; font-size: 10pt;">Application of Basic Skills in Games -</span></b><span style="font-family: Verdana, sans-serif; font-size: 10pt;">Students
participate in a variety of games to develop individual and manipulative
skills, techniques, strategies and spatial awareness. Inherent in playing all
games are cooperation, respect for others, fair play and etiquette.<o:p></o:p></span></div>
<div style="text-align: justify;">
<b><span style="font-family: Verdana, sans-serif; font-size: 10pt;">Application of Basic Skills in Types of Gymnastics -<span class="apple-converted-space"> </span></span></b><span style="font-family: Verdana, sans-serif; font-size: 10pt;">Students participate in
movement challenges that enable them to develop poise, grace, rhythm,
coordination, balance, strength, flexibility and effective body mechanics.<o:p></o:p></span></div>
<div style="text-align: justify;">
<b><span style="font-family: Verdana, sans-serif; font-size: 10pt;">Application of Basic Skills in Individual Activities -<span class="apple-converted-space"> </span></span></b><span style="font-family: Verdana, sans-serif; font-size: 10pt;">Students participate in
running, jumping and throwing activities; individual manipulatives; combative
or self-defense activities and target activities.<o:p></o:p></span></div>
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Locomotor and Nonlocomotor

Developing physical skills is a prerequisite to being active for life and physical education class is where it happens! Students need to...

NSTP 2

Nstp 2 from Tricia Andrea Lorena Pongyan

All about Polynomials

<span style="background-color: white; color: #333333; font-family: Verdana, Arial, sans-serif; font-size: 14px; line-height: 21px;">A </span><strong style="background-color: white; color: #333333; font-family: Verdana, Arial, sans-serif; font-size: 14px; line-height: 21px; margin: 0px; outline: 0px; padding: 0px;">polynomial</strong><span style="background-color: white; color: #333333; font-family: Verdana, Arial, sans-serif; font-size: 14px; line-height: 21px;"> consists of two or more terms. For example, </span><em style="background-color: white; color: #333333; font-family: Verdana, Arial, sans-serif; font-size: 14px; line-height: 21px; margin: 0px; outline: 0px; padding: 0px;">x</em><span style="background-color: white; color: #333333; font-family: Verdana, Arial, sans-serif; font-size: 14px; line-height: 21px;"> + </span><em style="background-color: white; color: #333333; font-family: Verdana, Arial, sans-serif; font-size: 14px; line-height: 21px; margin: 0px; outline: 0px; padding: 0px;">y</em><span style="background-color: white; color: #333333; font-family: Verdana, Arial, sans-serif; font-size: 14px; line-height: 21px;">, </span><em style="background-color: white; color: #333333; font-family: Verdana, Arial, sans-serif; font-size: 14px; line-height: 21px; margin: 0px; outline: 0px; padding: 0px;">y</em><sup style="background-color: white; color: #333333; font-family: Verdana, Arial, sans-serif; line-height: 21px; margin: 0px; outline: 0px; padding: 0px;">2</sup><span style="background-color: white; color: #333333; font-family: Verdana, Arial, sans-serif; font-size: 14px; line-height: 21px;"> – </span><em style="background-color: white; color: #333333; font-family: Verdana, Arial, sans-serif; font-size: 14px; line-height: 21px; margin: 0px; outline: 0px; padding: 0px;">x</em><sup style="background-color: white; color: #333333; font-family: Verdana, Arial, sans-serif; line-height: 21px; margin: 0px; outline: 0px; padding: 0px;">2</sup><span style="background-color: white; color: #333333; font-family: Verdana, Arial, sans-serif; font-size: 14px; line-height: 21px;">, and </span><em style="background-color: white; color: #333333; font-family: Verdana, Arial, sans-serif; font-size: 14px; line-height: 21px; margin: 0px; outline: 0px; padding: 0px;">x</em><sup style="background-color: white; color: #333333; font-family: Verdana, Arial, sans-serif; line-height: 21px; margin: 0px; outline: 0px; padding: 0px;">2</sup><span style="background-color: white; color: #333333; font-family: Verdana, Arial, sans-serif; font-size: 14px; line-height: 21px;"> + 3 </span><em style="background-color: white; color: #333333; font-family: Verdana, Arial, sans-serif; font-size: 14px; line-height: 21px; margin: 0px; outline: 0px; padding: 0px;">x</em><span style="background-color: white; color: #333333; font-family: Verdana, Arial, sans-serif; font-size: 14px; line-height: 21px;"> + 5 </span><em style="background-color: white; color: #333333; font-family: Verdana, Arial, sans-serif; font-size: 14px; line-height: 21px; margin: 0px; outline: 0px; padding: 0px;">y</em><sup style="background-color: white; color: #333333; font-family: Verdana, Arial, sans-serif; line-height: 21px; margin: 0px; outline: 0px; padding: 0px;">2</sup><span style="background-color: white; color: #333333; font-family: Verdana, Arial, sans-serif; font-size: 14px; line-height: 21px;"> are all polynomials. A </span><strong style="background-color: white; color: #333333; font-family: Verdana, Arial, sans-serif; font-size: 14px; line-height: 21px; margin: 0px; outline: 0px; padding: 0px;">binomial</strong><span style="background-color: white; color: #333333; font-family: Verdana, Arial, sans-serif; font-size: 14px; line-height: 21px;"> is a polynomial that consists of exactly two terms. For example, </span><em style="background-color: white; color: #333333; font-family: Verdana, Arial, sans-serif; font-size: 14px; line-height: 21px; margin: 0px; outline: 0px; padding: 0px;">x</em><span style="background-color: white; color: #333333; font-family: Verdana, Arial, sans-serif; font-size: 14px; line-height: 21px;"> + </span><em style="background-color: white; color: #333333; font-family: Verdana, Arial, sans-serif; font-size: 14px; line-height: 21px; margin: 0px; outline: 0px; padding: 0px;">y</em><span style="background-color: white; color: #333333; font-family: Verdana, Arial, sans-serif; font-size: 14px; line-height: 21px;"> is a binomial. A </span><strong style="background-color: white; color: #333333; font-family: Verdana, Arial, sans-serif; font-size: 14px; line-height: 21px; margin: 0px; outline: 0px; padding: 0px;">trinomial</strong><span style="background-color: white; color: #333333; font-family: Verdana, Arial, sans-serif; font-size: 14px; line-height: 21px;"> is a polynomial that consists of exactly three terms. For example, </span><em style="background-color: white; color: #333333; font-family: Verdana, Arial, sans-serif; font-size: 14px; line-height: 21px; margin: 0px; outline: 0px; padding: 0px;">y</em><sup style="background-color: white; color: #333333; font-family: Verdana, Arial, sans-serif; line-height: 21px; margin: 0px; outline: 0px; padding: 0px;">2</sup><span style="background-color: white; color: #333333; font-family: Verdana, Arial, sans-serif; font-size: 14px; line-height: 21px;"> + 9 </span><em style="background-color: white; color: #333333; font-family: Verdana, Arial, sans-serif; font-size: 14px; line-height: 21px; margin: 0px; outline: 0px; padding: 0px;">y</em><span style="background-color: white; color: #333333; font-family: Verdana, Arial, sans-serif; font-size: 14px; line-height: 21px;"> + 8 is a trinomial.</span><br />
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<span style="background-color: white; color: #333333; font-family: Verdana, Arial, sans-serif; font-size: 14px; line-height: 21px;">Polynomials usually are arranged in one of two ways. </span><strong style="background-color: white; color: #333333; font-family: Verdana, Arial, sans-serif; font-size: 14px; line-height: 21px; margin: 0px; outline: 0px; padding: 0px;">Ascending order</strong><span style="background-color: white; color: #333333; font-family: Verdana, Arial, sans-serif; font-size: 14px; line-height: 21px;"> is basically when the power of a term increases for each succeeding term. For example, </span><em style="background-color: white; color: #333333; font-family: Verdana, Arial, sans-serif; font-size: 14px; line-height: 21px; margin: 0px; outline: 0px; padding: 0px;">x</em><span style="background-color: white; color: #333333; font-family: Verdana, Arial, sans-serif; font-size: 14px; line-height: 21px;"> + </span><em style="background-color: white; color: #333333; font-family: Verdana, Arial, sans-serif; font-size: 14px; line-height: 21px; margin: 0px; outline: 0px; padding: 0px;">x</em><sup style="background-color: white; color: #333333; font-family: Verdana, Arial, sans-serif; line-height: 21px; margin: 0px; outline: 0px; padding: 0px;">2</sup><span style="background-color: white; color: #333333; font-family: Verdana, Arial, sans-serif; font-size: 14px; line-height: 21px;"> + </span><em style="background-color: white; color: #333333; font-family: Verdana, Arial, sans-serif; font-size: 14px; line-height: 21px; margin: 0px; outline: 0px; padding: 0px;">x</em><sup style="background-color: white; color: #333333; font-family: Verdana, Arial, sans-serif; line-height: 21px; margin: 0px; outline: 0px; padding: 0px;">3</sup><span style="background-color: white; color: #333333; font-family: Verdana, Arial, sans-serif; font-size: 14px; line-height: 21px;"> or 5 </span><em style="background-color: white; color: #333333; font-family: Verdana, Arial, sans-serif; font-size: 14px; line-height: 21px; margin: 0px; outline: 0px; padding: 0px;">x</em><span style="background-color: white; color: #333333; font-family: Verdana, Arial, sans-serif; font-size: 14px; line-height: 21px;"> + 2 </span><em style="background-color: white; color: #333333; font-family: Verdana, Arial, sans-serif; font-size: 14px; line-height: 21px; margin: 0px; outline: 0px; padding: 0px;">x</em><sup style="background-color: white; color: #333333; font-family: Verdana, Arial, sans-serif; line-height: 21px; margin: 0px; outline: 0px; padding: 0px;">2</sup><span style="background-color: white; color: #333333; font-family: Verdana, Arial, sans-serif; font-size: 14px; line-height: 21px;">– 3 </span><em style="background-color: white; color: #333333; font-family: Verdana, Arial, sans-serif; font-size: 14px; line-height: 21px; margin: 0px; outline: 0px; padding: 0px;">x</em><sup style="background-color: white; color: #333333; font-family: Verdana, Arial, sans-serif; line-height: 21px; margin: 0px; outline: 0px; padding: 0px;">3</sup><span style="background-color: white; color: #333333; font-family: Verdana, Arial, sans-serif; font-size: 14px; line-height: 21px;"> + </span><em style="background-color: white; color: #333333; font-family: Verdana, Arial, sans-serif; font-size: 14px; line-height: 21px; margin: 0px; outline: 0px; padding: 0px;">x</em><sup style="background-color: white; color: #333333; font-family: Verdana, Arial, sans-serif; line-height: 21px; margin: 0px; outline: 0px; padding: 0px;">5</sup><span style="background-color: white; color: #333333; font-family: Verdana, Arial, sans-serif; font-size: 14px; line-height: 21px;"> are arranged in ascending order. </span><strong style="background-color: white; color: #333333; font-family: Verdana, Arial, sans-serif; font-size: 14px; line-height: 21px; margin: 0px; outline: 0px; padding: 0px;">Descending order</strong><span style="background-color: white; color: #333333; font-family: Verdana, Arial, sans-serif; font-size: 14px; line-height: 21px;"> is basically when the power of a term decreases for each succeeding term. For example, </span><em style="background-color: white; color: #333333; font-family: Verdana, Arial, sans-serif; font-size: 14px; line-height: 21px; margin: 0px; outline: 0px; padding: 0px;">x</em><sup style="background-color: white; color: #333333; font-family: Verdana, Arial, sans-serif; line-height: 21px; margin: 0px; outline: 0px; padding: 0px;">3</sup><span style="background-color: white; color: #333333; font-family: Verdana, Arial, sans-serif; font-size: 14px; line-height: 21px;"> + </span><em style="background-color: white; color: #333333; font-family: Verdana, Arial, sans-serif; font-size: 14px; line-height: 21px; margin: 0px; outline: 0px; padding: 0px;">x</em><sup style="background-color: white; color: #333333; font-family: Verdana, Arial, sans-serif; line-height: 21px; margin: 0px; outline: 0px; padding: 0px;">2</sup><span style="background-color: white; color: #333333; font-family: Verdana, Arial, sans-serif; font-size: 14px; line-height: 21px;"> + </span><em style="background-color: white; color: #333333; font-family: Verdana, Arial, sans-serif; font-size: 14px; line-height: 21px; margin: 0px; outline: 0px; padding: 0px;">x</em><span style="background-color: white; color: #333333; font-family: Verdana, Arial, sans-serif; font-size: 14px; line-height: 21px;"> or 2 </span><em style="background-color: white; color: #333333; font-family: Verdana, Arial, sans-serif; font-size: 14px; line-height: 21px; margin: 0px; outline: 0px; padding: 0px;">x</em><sup style="background-color: white; color: #333333; font-family: Verdana, Arial, sans-serif; line-height: 21px; margin: 0px; outline: 0px; padding: 0px;">4</sup><span style="background-color: white; color: #333333; font-family: Verdana, Arial, sans-serif; font-size: 14px; line-height: 21px;"> + 3 </span><em style="background-color: white; color: #333333; font-family: Verdana, Arial, sans-serif; font-size: 14px; line-height: 21px; margin: 0px; outline: 0px; padding: 0px;">x</em><sup style="background-color: white; color: #333333; font-family: Verdana, Arial, sans-serif; line-height: 21px; margin: 0px; outline: 0px; padding: 0px;">2</sup><span style="background-color: white; color: #333333; font-family: Verdana, Arial, sans-serif; font-size: 14px; line-height: 21px;"> + 7 </span><em style="background-color: white; color: #333333; font-family: Verdana, Arial, sans-serif; font-size: 14px; line-height: 21px; margin: 0px; outline: 0px; padding: 0px;">x</em><span style="background-color: white; color: #333333; font-family: Verdana, Arial, sans-serif; font-size: 14px; line-height: 21px;"> are arranged in descending order. Descending order is more commonly used.</span><br />
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Adding and subtracting polynomials</h4>
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To <em style="margin: 0px; outline: 0px; padding: 0px;">add</em> or <em style="margin: 0px; outline: 0px; padding: 0px;">subtract polynomials</em>, just arrange <em style="margin: 0px; outline: 0px; padding: 0px;">like terms</em> in columns and then add or subtract. (Or simply add or subtract like terms when rearrangement is not necessary.)</div>
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Example 1</h5>
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Do the indicated arithmetic.</div>
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Add the polynomials.</div>
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Subtract the polynomials.</div>
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Multiplying polynomials</h4>
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To <em style="margin: 0px; outline: 0px; padding: 0px;">multiply polynomials,</em> multiply each term in one polynomial by each term in the other polynomial. Then simplify if necessary.</div>
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Example 2</h5>
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Multiply.</div>
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<displayeditem numbered="no" style="margin: 0px; outline: 0px; padding: 0px;" type="mathematics" xml:id="c06-disp-0007"><img alt="equation" src="http://www.cliffsnotes.com/assets/253652.png" style="border: 0px; margin: 0px; outline: 0px; padding: 0px;" /></displayeditem></div>
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Or you may want to use the “ <strong style="margin: 0px; outline: 0px; padding: 0px;">F.O.I.L.</strong>” method with <em style="margin: 0px; outline: 0px; padding: 0px;">binomials.</em> <strong style="margin: 0px; outline: 0px; padding: 0px;">F.O.I.L.</strong> means<strong style="margin: 0px; outline: 0px; padding: 0px;">F</strong>irst terms, <strong style="margin: 0px; outline: 0px; padding: 0px;">O</strong>utside terms, <strong style="margin: 0px; outline: 0px; padding: 0px;">I</strong>nside terms, <strong style="margin: 0px; outline: 0px; padding: 0px;">L</strong>ast terms. Then simplify if necessary.</div>
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Example 3</h5>
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Multiply.</div>
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(3 <em style="margin: 0px; outline: 0px; padding: 0px;">x</em> + <em style="margin: 0px; outline: 0px; padding: 0px;">a</em>)(2 <em style="margin: 0px; outline: 0px; padding: 0px;">x</em> – 2 <em style="margin: 0px; outline: 0px; padding: 0px;">a</em>) =</div>
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Multiply <em style="margin: 0px; outline: 0px; padding: 0px;">first</em> terms from each quantity.</div>
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Now <em style="margin: 0px; outline: 0px; padding: 0px;">outside</em> terms.</div>
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Now <em style="margin: 0px; outline: 0px; padding: 0px;">inside</em> terms.</div>
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Finally <em style="margin: 0px; outline: 0px; padding: 0px;">last</em> terms.</div>
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Now simplify.</div>
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6 <em style="margin: 0px; outline: 0px; padding: 0px;">x</em><sup style="margin: 0px; outline: 0px; padding: 0px;">2</sup> – 6 <em style="margin: 0px; outline: 0px; padding: 0px;">ax</em> + 2 <em style="margin: 0px; outline: 0px; padding: 0px;">ax</em> – 2 <em style="margin: 0px; outline: 0px; padding: 0px;">a</em><sup style="margin: 0px; outline: 0px; padding: 0px;">2</sup> = 6 <em style="margin: 0px; outline: 0px; padding: 0px;">x</em><sup style="margin: 0px; outline: 0px; padding: 0px;">2</sup> – 4 <em style="margin: 0px; outline: 0px; padding: 0px;">ax</em> – 2 <em style="margin: 0px; outline: 0px; padding: 0px;">a</em><sup style="margin: 0px; outline: 0px; padding: 0px;">2</sup></div>
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Example 4</h5>
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Multiply.</div>
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This operation also can be done using the distributive property.</div>
<div style="margin-bottom: 1em; margin-top: 1em; outline: 0px; padding: 0px;">
<displayeditem numbered="no" style="margin: 0px; outline: 0px; padding: 0px;" type="mathematics" xml:id="c06-disp-0015"><img alt="equation" src="http://www.cliffsnotes.com/assets/253688.png" style="border: 0px; margin: 0px; outline: 0px; padding: 0px;" /></displayeditem></div>
<h4 style="font-size: 17px; margin: 20px 0px 0px; outline: 0px; padding: 0px;">
Dividing polynomials by monomials</h4>
<section style="margin: 0px; outline: 0px; padding: 0px;" xml:id="c06-sec-0010"><div style="margin-bottom: 1em; margin-top: 1em; outline: 0px; padding: 0px;">
To <em style="margin: 0px; outline: 0px; padding: 0px;">divide a polynomial by a monomial,</em> just divide each term in the polynomial by the monomial.</div>
<h5 style="font-size: 14px; margin: 15px 0px 0px; outline: 0px; padding: 0px;">
Example 5</h5>
<div style="margin-bottom: 1em; margin-top: 1em; outline: 0px; padding: 0px;">
<mathstatement style="margin: 0px; outline: 0px; padding: 0px;" xml:id="c06-mthst-0010"></mathstatement></div>
<div style="margin-bottom: 1em; margin-top: 1em; outline: 0px; padding: 0px;">
Divide.</div>
<div style="margin-bottom: 1em; margin-top: 1em; outline: 0px; padding: 0px;">
<displayeditem numbered="no" style="margin: 0px; outline: 0px; padding: 0px;" type="mathematics" xml:id="c06-disp-0016"><img alt="equation" src="http://www.cliffsnotes.com/assets/253658.png" style="border: 0px; margin: 0px; outline: 0px; padding: 0px;" /></displayeditem></div>
<div style="margin-bottom: 1em; margin-top: 1em; outline: 0px; padding: 0px;">
<displayeditem numbered="no" style="margin: 0px; outline: 0px; padding: 0px;" type="mathematics" xml:id="c06-disp-0017"><img alt="equation" src="http://www.cliffsnotes.com/assets/253659.png" style="border: 0px; margin: 0px; outline: 0px; padding: 0px;" /></displayeditem></div>
<h4 style="font-size: 17px; margin: 20px 0px 0px; outline: 0px; padding: 0px;">
Dividing polynomials by polynomials</h4>
<section style="margin: 0px; outline: 0px; padding: 0px;" xml:id="c06-sec-0011"><div style="margin-bottom: 1em; margin-top: 1em; outline: 0px; padding: 0px;">
To <em style="margin: 0px; outline: 0px; padding: 0px;">divide a polynomial by a polynomial,</em> make sure both are in descending order; then use long division. ( <em style="margin: 0px; outline: 0px; padding: 0px;">Remember:</em> Divide by the first term, multiply, subtract, bring down.)</div>
<h5 style="font-size: 14px; margin: 15px 0px 0px; outline: 0px; padding: 0px;">
Example 6</h5>
<div style="margin-bottom: 1em; margin-top: 1em; outline: 0px; padding: 0px;">
<mathstatement style="margin: 0px; outline: 0px; padding: 0px;" xml:id="c06-mthst-0011"></mathstatement></div>
<div style="margin-bottom: 1em; margin-top: 1em; outline: 0px; padding: 0px;">
Divide 4 <em style="margin: 0px; outline: 0px; padding: 0px;">a</em><sup style="margin: 0px; outline: 0px; padding: 0px;">2</sup> + 18 <em style="margin: 0px; outline: 0px; padding: 0px;">a</em> + 8 by <em style="margin: 0px; outline: 0px; padding: 0px;">a</em> + 4.</div>
<div style="margin-bottom: 1em; margin-top: 1em; outline: 0px; padding: 0px;">
<displayeditem numbered="no" style="margin: 0px; outline: 0px; padding: 0px;" type="mathematics" xml:id="c06-disp-0018"><img alt="equation" src="http://www.cliffsnotes.com/assets/253660.png" style="border: 0px; margin: 0px; outline: 0px; padding: 0px;" /></displayeditem></div>
<div style="margin-bottom: 1em; margin-top: 1em; outline: 0px; padding: 0px;">
<displayeditem numbered="no" style="margin: 0px; outline: 0px; padding: 0px;" type="mathematics" xml:id="c06-disp-0019"><img alt="equation" src="http://www.cliffsnotes.com/assets/253661.png" style="border: 0px; margin: 0px; outline: 0px; padding: 0px;" /></displayeditem></div>
<div style="margin-bottom: 1em; margin-top: 1em; outline: 0px; padding: 0px;">
<displayeditem numbered="no" style="margin: 0px; outline: 0px; padding: 0px;" type="mathematics" xml:id="c06-disp-0020"><img alt="equation" src="http://www.cliffsnotes.com/assets/253662.png" style="border: 0px; margin: 0px; outline: 0px; padding: 0px;" /></displayeditem></div>
<div style="margin-bottom: 1em; margin-top: 1em; outline: 0px; padding: 0px;">
<displayeditem numbered="no" style="margin: 0px; outline: 0px; padding: 0px;" type="mathematics" xml:id="c06-disp-0021"><img alt="equation" src="http://www.cliffsnotes.com/assets/253663.png" style="border: 0px; margin: 0px; outline: 0px; padding: 0px;" /></displayeditem></div>
<div style="margin-bottom: 1em; margin-top: 1em; outline: 0px; padding: 0px;">
<displayeditem numbered="no" style="margin: 0px; outline: 0px; padding: 0px;" type="mathematics" xml:id="c06-disp-0022"><img alt="equation" src="http://www.cliffsnotes.com/assets/253664.png" style="border: 0px; margin: 0px; outline: 0px; padding: 0px;" /></displayeditem></div>
<div style="margin-bottom: 1em; margin-top: 1em; outline: 0px; padding: 0px;">
<displayeditem numbered="no" style="margin: 0px; outline: 0px; padding: 0px;" type="mathematics" xml:id="c06-disp-0025"><img alt="equation" src="http://www.cliffsnotes.com/assets/253665.png" style="border: 0px; margin: 0px; outline: 0px; padding: 0px;" /></displayeditem></div>
<div style="margin-bottom: 1em; margin-top: 1em; outline: 0px; padding: 0px;">
<displayeditem numbered="no" style="margin: 0px; outline: 0px; padding: 0px;" type="mathematics" xml:id="c06-disp-0026"><img alt="equation" src="http://www.cliffsnotes.com/assets/253666.png" style="border: 0px; margin: 0px; outline: 0px; padding: 0px;" /></displayeditem></div>
<div style="margin-bottom: 1em; margin-top: 1em; outline: 0px; padding: 0px;">
<displayeditem numbered="no" style="margin: 0px; outline: 0px; padding: 0px;" type="mathematics" xml:id="c06-disp-0027"><img alt="equation" src="http://www.cliffsnotes.com/assets/253667.png" style="border: 0px; margin: 0px; outline: 0px; padding: 0px;" /></displayeditem></div>
<h5 style="font-size: 14px; margin: 15px 0px 0px; outline: 0px; padding: 0px;">
Example 7</h5>
<div style="margin-bottom: 1em; margin-top: 1em; outline: 0px; padding: 0px;">
<mathstatement style="margin: 0px; outline: 0px; padding: 0px;" xml:id="c06-mthst-0012"></mathstatement></div>
<div style="margin-bottom: 1em; margin-top: 1em; outline: 0px; padding: 0px;">
Divide.</div>
<ol style="margin: 1em 0px; outline: 0px; padding: 0px;">
<li style="margin: 0px 0px 0px 40px; outline: 0px; padding: 0px;"><div style="margin-bottom: 1em; margin-top: 1em; outline: 0px; padding: 0px;">
<img alt="equation" src="http://www.cliffsnotes.com/assets/253706.png" style="border: 0px; margin: 0px; outline: 0px; padding: 0px;" /></div>
<div style="margin-bottom: 1em; margin-top: 1em; outline: 0px; padding: 0px;">
<displayeditem numbered="no" style="margin: 0px; outline: 0px; padding: 0px;" type="mathematics" xml:id="c06-disp-0028"><img alt="equation" src="http://www.cliffsnotes.com/assets/253707.png" style="border: 0px; margin: 0px; outline: 0px; padding: 0px;" /></displayeditem></div>
</li>
<li style="margin: 0px 0px 0px 40px; outline: 0px; padding: 0px;"><div style="margin-bottom: 1em; margin-top: 1em; outline: 0px; padding: 0px;">
<img alt="equation" src="http://www.cliffsnotes.com/assets/253704.png" style="border: 0px; margin: 0px; outline: 0px; padding: 0px;" /></div>
<div style="margin-bottom: 1em; margin-top: 1em; outline: 0px; padding: 0px;">
First change to descending order: <em style="margin: 0px; outline: 0px; padding: 0px;">x</em><sup style="margin: 0px; outline: 0px; padding: 0px;">2</sup> + 2 <em style="margin: 0px; outline: 0px; padding: 0px;">x</em> + 1. Then divide.</div>
<div style="margin-bottom: 1em; margin-top: 1em; outline: 0px; padding: 0px;">
<displayeditem numbered="no" style="margin: 0px; outline: 0px; padding: 0px;" type="mathematics" xml:id="c06-disp-0029"><img alt="equation" src="http://www.cliffsnotes.com/assets/253705.png" style="border: 0px; margin: 0px; outline: 0px; padding: 0px;" /></displayeditem></div>
</li>
<li style="margin: 0px 0px 0px 40px; outline: 0px; padding: 0px;"><div style="margin-bottom: 1em; margin-top: 1em; outline: 0px; padding: 0px;">
<img alt="equation" src="http://www.cliffsnotes.com/assets/253700.png" style="border: 0px; margin: 0px; outline: 0px; padding: 0px;" /></div>
<div style="margin-bottom: 1em; margin-top: 1em; outline: 0px; padding: 0px;">
Note: When terms are missing, be sure to leave proper room between terms.</div>
<div style="margin-bottom: 1em; margin-top: 1em; outline: 0px; padding: 0px;">
<displayeditem numbered="no" style="margin: 0px; outline: 0px; padding: 0px;" type="mathematics" xml:id="c06-disp-0030"><img alt="equation" src="http://www.cliffsnotes.com/assets/253701.png" style="border: 0px; margin: 0px; outline: 0px; padding: 0px;" /></displayeditem></div>
</li>
<li style="margin: 0px 0px 0px 40px; outline: 0px; padding: 0px;"><div style="margin-bottom: 1em; margin-top: 1em; outline: 0px; padding: 0px;">
<img alt="equation" src="http://www.cliffsnotes.com/assets/253708.png" style="border: 0px; margin: 0px; outline: 0px; padding: 0px;" /></div>
<div style="margin-bottom: 1em; margin-top: 1em; outline: 0px; padding: 0px;">
<displayeditem numbered="no" style="margin: 0px; outline: 0px; padding: 0px;" type="mathematics" xml:id="c06-disp-0031"><img alt="equation" src="http://www.cliffsnotes.com/assets/253709.png" style="border: 0px; margin: 0px; outline: 0px; padding: 0px;" /></displayeditem></div>
</li>
<li style="margin: 0px 0px 0px 40px; outline: 0px; padding: 0px;"><div style="margin-bottom: 1em; margin-top: 1em; outline: 0px; padding: 0px;">
<img alt="equation" src="http://www.cliffsnotes.com/assets/253671.png" style="border: 0px; margin: 0px; outline: 0px; padding: 0px;" /></div>
<div style="margin-bottom: 1em; margin-top: 1em; outline: 0px; padding: 0px;">
This answer can be rewritten as <img alt="equation" src="http://www.cliffsnotes.com/assets/253672.png" style="border: 0px; margin: 0px; outline: 0px; padding: 0px;" /></div>
</li>
</ol>
</section></section></section></section>

All about Polynomials

A polynomial consists of two or more terms. For example, x + y , y 2 – x 2 , and x 2 + 3 x + 5 y 2 are all polynomials. A bin...

History of Humanities

<div class="separator" style="clear: both; text-align: center;">
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhQD4wPSDl_mRW3lRRWO71xst3sOrt1hpJn_kJ9o5E49ufRbineUey4dsdxBSm5GR0g24gW4JlSdiNKihmhMllq_Gz0gIR9yBeVPo2WcghzKHbhBymmuzg0r32usnKscfmaTih1aoXWAhNF/s1600/1.jpeg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhQD4wPSDl_mRW3lRRWO71xst3sOrt1hpJn_kJ9o5E49ufRbineUey4dsdxBSm5GR0g24gW4JlSdiNKihmhMllq_Gz0gIR9yBeVPo2WcghzKHbhBymmuzg0r32usnKscfmaTih1aoXWAhNF/s1600/1.jpeg" /></a></div>
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<b><span style="font-family: Arial, sans-serif; font-size: 10.5pt;"><br /></span></b></div>
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<b><span style="font-family: Arial, sans-serif; font-size: 10.5pt;">HUMANITIES</span></b><span class="apple-converted-space"><span style="font-family: Arial, sans-serif; font-size: 10.5pt;"> </span></span><span style="font-family: Arial, sans-serif; font-size: 10.5pt;">are<span class="apple-converted-space"> </span><span style="color: black; mso-themecolor: text1; text-decoration: none; text-underline: none;">academic disciplines</span><span class="apple-converted-space"> </span>that study human culture. The
humanities use methods that are primarily<span class="apple-converted-space"> </span><span style="color: black; mso-themecolor: text1; text-decoration: none; text-underline: none;">critical</span>, or<span class="apple-converted-space"> </span><span style="color: black; mso-themecolor: text1; text-decoration: none; text-underline: none;">speculative</span>, and
have a significant historical element</span><sup id="cite_ref-oed_1-0" style="unicode-bidi: -webkit-isolate;"><span style="font-family: Arial, sans-serif; font-size: 8.5pt;"><a href="http://en.wikipedia.org/wiki/Humanities#cite_note-oed-1"><span style="color: black; mso-themecolor: text1; text-decoration: none; text-underline: none;">[1]</span></a></span></sup><span style="font-family: Arial, sans-serif; font-size: 10.5pt;">—as distinguished from
the mainly<span class="apple-converted-space"> </span><span style="color: black; mso-themecolor: text1; text-decoration: none; text-underline: none;">empirical</span><span class="apple-converted-space"> </span>approaches
of the<span class="apple-converted-space"> </span><span style="color: black; mso-themecolor: text1; text-decoration: none; text-underline: none;">natural sciences</span>.</span><sup id="cite_ref-oed_1-1" style="unicode-bidi: -webkit-isolate;"><span style="font-family: Arial, sans-serif; font-size: 8.5pt;"><a href="http://en.wikipedia.org/wiki/Humanities#cite_note-oed-1"><span style="color: black; mso-themecolor: text1; text-decoration: none; text-underline: none;">[1]</span></a></span></sup><span class="apple-converted-space"><span style="font-family: Arial, sans-serif; font-size: 10.5pt;"> </span></span><span style="font-family: Arial, sans-serif; font-size: 10.5pt;">The
humanities include<span class="apple-converted-space"> </span><span style="color: black; mso-themecolor: text1; text-decoration: none; text-underline: none;">ancient and
modern languages</span>,<span class="apple-converted-space"> </span><span style="color: black; mso-themecolor: text1; text-decoration: none; text-underline: none;">literature</span>,<span class="apple-converted-space"> </span><span style="color: black; mso-themecolor: text1; text-decoration: none; text-underline: none;">philosophy</span>,<span class="apple-converted-space"> </span><span style="color: black; mso-themecolor: text1; text-decoration: none; text-underline: none;">religion</span>,
and<span class="apple-converted-space"> </span><span style="color: black; mso-themecolor: text1; text-decoration: none; text-underline: none;">visual</span><span class="apple-converted-space"> </span>and<span class="apple-converted-space"> </span><span style="color: black; mso-themecolor: text1; text-decoration: none; text-underline: none;">performing arts</span><span class="apple-converted-space"> </span>such
as<span class="apple-converted-space"> </span><span style="color: black; mso-themecolor: text1; text-decoration: none; text-underline: none;">music</span><span class="apple-converted-space"> </span>and <span style="color: black; mso-themecolor: text1; text-decoration: none; text-underline: none;">theatre  </span>.
Areas that are sometimes regarded</span><sup><span style="font-family: Arial, sans-serif; font-size: 8.5pt;">[<i><a href="http://en.wikipedia.org/wiki/Wikipedia:Manual_of_Style/Words_to_watch#Unsupported_attributions" title="Wikipedia:Manual of Style/Words to watch"><span style="color: black; mso-themecolor: text1; text-decoration: none; text-underline: none;"><span title="The material near this tag may use weasel words or too-vague attribution. (July 2014)">by
whom?</span></span></a></i>]</span></sup><span class="apple-converted-space"><span style="font-family: Arial, sans-serif; font-size: 10.5pt;"> </span></span><span style="font-family: Arial, sans-serif; font-size: 10.5pt;">as<span class="apple-converted-space"> </span><span style="color: black; mso-themecolor: text1; text-decoration: none; text-underline: none;">social sciences</span><span class="apple-converted-space"> </span>and
sometimes as humanities include<span class="apple-converted-space"> </span><span style="color: black; mso-themecolor: text1; text-decoration: none; text-underline: none;">history</span>,<span class="apple-converted-space"> </span><span style="color: black; mso-themecolor: text1; text-decoration: none; text-underline: none;">archaeology</span>,<span style="color: black; mso-themecolor: text1; text-decoration: none; text-underline: none;">anthropology</span>,<span class="apple-converted-space"> </span><span style="color: black; mso-themecolor: text1; text-decoration: none; text-underline: none;">area studies</span>,<span class="apple-converted-space"> </span><span style="color: black; mso-themecolor: text1; text-decoration: none; text-underline: none;">communication studies</span>,<span class="apple-converted-space"> </span><span style="color: black; mso-themecolor: text1; text-decoration: none; text-underline: none;">classical studies</span>,<span class="apple-converted-space"> </span><span style="color: black; mso-themecolor: text1; text-decoration: none; text-underline: none;">law</span>,<span class="apple-converted-space"> </span><span style="color: black; mso-themecolor: text1; text-decoration: none; text-underline: none;">semiotics</span><span class="apple-converted-space"> </span>and<span class="apple-converted-space"> </span><span style="color: black; mso-themecolor: text1; text-decoration: none; text-underline: none;">linguistics</span>.<o:p></o:p></span></div>
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<span style="font-family: Arial, sans-serif; font-size: 10.5pt;">Scholars in the humanities are "humanities
scholars" or<span class="apple-converted-space"> </span><i>humanists</i>.</span><sup id="cite_ref-2" style="unicode-bidi: -webkit-isolate;"><span style="font-family: Arial, sans-serif; font-size: 8.5pt;"><a href="http://en.wikipedia.org/wiki/Humanities#cite_note-2"><span style="color: black; mso-themecolor: text1; text-decoration: none; text-underline: none;">[2]</span></a></span></sup><span class="apple-converted-space"><span style="font-family: Arial, sans-serif; font-size: 10.5pt;"> </span></span><span style="font-family: Arial, sans-serif; font-size: 10.5pt;">The
term "humanist" also describes the philosophical position of<span style="color: black; mso-themecolor: text1; text-decoration: none; text-underline: none;">humanism</span>,
which some "<span style="color: black; mso-themecolor: text1; text-decoration: none; text-underline: none;">antihumanist</span>" scholars in the
humanities refuse. The Renaissance scholars and artists were also called<span class="apple-converted-space"> </span><span style="color: black; mso-themecolor: text1; text-decoration: none; text-underline: none;">humanists</span>. Some<span class="apple-converted-space"> </span><span style="color: black; mso-themecolor: text1; text-decoration: none; text-underline: none;">secondary schools</span><span class="apple-converted-space"> </span>offer
humanities classes, usually consisting of<span class="apple-converted-space"> </span><span style="color: black; mso-themecolor: text1; text-decoration: none; text-underline: none;">English</span><span class="apple-converted-space"> </span><span style="color: black; mso-themecolor: text1; text-decoration: none; text-underline: none;">literature</span>, global studies, and<span class="apple-converted-space"> </span><span style="color: black; mso-themecolor: text1; text-decoration: none; text-underline: none;">art</span>.<o:p></o:p></span></div>
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<span style="font-family: Arial, sans-serif; font-size: 10.5pt;">Human disciplines like<span class="apple-converted-space"> </span><span style="color: black; mso-themecolor: text1; text-decoration: none; text-underline: none;">history</span><span class="apple-converted-space"> </span>and<span class="apple-converted-space"> </span><span style="color: black; mso-themecolor: text1; text-decoration: none; text-underline: none;">cultural anthropology</span><span class="apple-converted-space"> </span>study subject matters that the<span class="apple-converted-space"> </span><span style="color: black; mso-themecolor: text1; text-decoration: none; text-underline: none;">experimental method</span><span class="apple-converted-space"> </span>does not apply to—and instead mainly
use the<span class="apple-converted-space"> </span><span style="color: black; mso-themecolor: text1; text-decoration: none; text-underline: none;">comparative method</span></span><sup id="cite_ref-Wallace2008p28_3-0" style="unicode-bidi: -webkit-isolate;"><span style="font-family: Arial, sans-serif; font-size: 8.5pt;"><a href="http://en.wikipedia.org/wiki/Humanities#cite_note-Wallace2008p28-3"><span style="color: black; mso-themecolor: text1; text-decoration: none; text-underline: none;">[3]</span></a></span></sup><span class="apple-converted-space"><span style="font-family: Arial, sans-serif; font-size: 10.5pt;"> </span></span><span style="font-family: Arial, sans-serif; font-size: 10.5pt;">and<span class="apple-converted-space"> </span><span style="color: black; mso-themecolor: text1; text-decoration: none; text-underline: none;">comparative research</span>.<o:p></o:p></span></div>
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<b><span style="font-family: Arial, sans-serif; font-size: 14pt;">History</span></b><b><span style="font-family: Arial, sans-serif; font-size: 14pt;"><o:p></o:p></span></b></div>
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<span style="font-family: Arial, sans-serif; font-size: 10.5pt;"><span style="color: black; mso-bidi-font-size: 11.0pt; mso-themecolor: text1; text-decoration: none; text-underline: none;">History</span></span><span style="font-family: Arial, sans-serif; font-size: 10.5pt;"> </span><span style="font-family: Arial, sans-serif; font-size: 10.5pt;">is systematically collected</span><span style="font-family: Arial, sans-serif; font-size: 10.5pt;"> </span><span style="font-family: Arial, sans-serif; font-size: 10.5pt;"><span style="color: black; mso-bidi-font-size: 11.0pt; mso-themecolor: text1; text-decoration: none; text-underline: none;">information</span></span><span style="font-family: Arial, sans-serif; font-size: 10.5pt;"> </span><span style="font-family: Arial, sans-serif; font-size: 10.5pt;">about the</span><span style="font-family: Arial, sans-serif; font-size: 10.5pt;"> </span><span style="font-family: Arial, sans-serif; font-size: 10.5pt;"><span style="color: black; mso-bidi-font-size: 11.0pt; mso-themecolor: text1; text-decoration: none; text-underline: none;">past</span>. When used as the name of a</span><span style="font-family: Arial, sans-serif; font-size: 10.5pt;"> </span><span style="font-family: Arial, sans-serif; font-size: 10.5pt;"><span style="color: black; mso-bidi-font-size: 11.0pt; mso-themecolor: text1; text-decoration: none; text-underline: none;">field of study</span>,</span><span style="font-family: Arial, sans-serif; font-size: 10.5pt;"> </span><i><span style="font-family: Arial, sans-serif; font-size: 10.5pt;">history</span></i><span style="font-family: Arial, sans-serif; font-size: 10.5pt;"> </span><span style="font-family: Arial, sans-serif; font-size: 10.5pt;">refers to the study and
interpretation of the record of</span><span style="font-family: Arial, sans-serif; font-size: 10.5pt;"> </span><span style="font-family: Arial, sans-serif; font-size: 10.5pt;"><span style="color: black; mso-bidi-font-size: 11.0pt; mso-themecolor: text1; text-decoration: none; text-underline: none;">humans</span>,</span><span style="font-family: Arial, sans-serif; font-size: 10.5pt;"> </span><span style="font-family: Arial, sans-serif; font-size: 10.5pt;"><span style="color: black; mso-bidi-font-size: 11.0pt; mso-themecolor: text1; text-decoration: none; text-underline: none;">societies</span>, institutions, and any topic
that has changed over time.<o:p></o:p></span></div>
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<span style="font-family: Arial, sans-serif; font-size: 10.5pt;">Traditionally, the study of history has been
considered a part of the humanities. In modern</span><span style="font-family: Arial, sans-serif; font-size: 10.5pt;"> </span><span style="font-family: Arial, sans-serif; font-size: 10.5pt;"><span style="color: black; mso-bidi-font-size: 11.0pt; mso-themecolor: text1; text-decoration: none; text-underline: none;">academia</span>, history is occasionally
classified as a<span style="color: black; mso-bidi-font-size: 11.0pt; mso-themecolor: text1; text-decoration: none; text-underline: none;">social science</span>.<o:p></o:p></span></div>
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<b><span style="font-family: Arial, sans-serif; font-size: 10.5pt;">History</span></b><span class="apple-converted-space"><span style="font-family: Arial, sans-serif; font-size: 10.5pt;"> </span></span><span style="font-family: Arial, sans-serif; font-size: 10.5pt;">(from<span class="apple-converted-space"> </span><span style="color: black; mso-themecolor: text1; text-decoration: none; text-underline: none;">Greek</span><span class="apple-converted-space"> </span></span><span xml:lang="el"><span lang="EL" style="font-family: Arial, sans-serif; font-size: 10.5pt;">ἱστορία</span></span><span style="font-family: Arial, sans-serif; font-size: 10.5pt;">,<span class="apple-converted-space"> </span><i>historia</i>,
meaning "inquiry, knowledge acquired by investigation")</span><sup id="cite_ref-JosephJanda_2-0" style="unicode-bidi: -webkit-isolate;"><span style="font-family: Arial, sans-serif; font-size: 8.5pt;"><a href="http://en.wikipedia.org/wiki/History#cite_note-JosephJanda-2"><span style="color: black; mso-themecolor: text1; text-decoration: none; text-underline: none;">[2]</span></a></span></sup><span class="apple-converted-space"><span style="font-family: Arial, sans-serif; font-size: 10.5pt;"> </span></span><span style="font-family: Arial, sans-serif; font-size: 10.5pt;">is the study
of the past, particularly how it relates to humans.</span><sup id="cite_ref-3" style="unicode-bidi: -webkit-isolate;"><span style="font-family: Arial, sans-serif; font-size: 8.5pt;"><a href="http://en.wikipedia.org/wiki/History#cite_note-3"><span style="color: black; mso-themecolor: text1; text-decoration: none; text-underline: none;">[3]</span></a><a href="http://en.wikipedia.org/wiki/History#cite_note-4"><span style="color: black; mso-themecolor: text1; text-decoration: none; text-underline: none;">[4]</span></a></span></sup><span class="apple-converted-space"><span style="font-family: Arial, sans-serif; font-size: 10.5pt;"> </span></span><span style="font-family: Arial, sans-serif; font-size: 10.5pt;">It is
an<span class="apple-converted-space"> </span><span style="color: black; mso-themecolor: text1; text-decoration: none; text-underline: none;">umbrella term</span><span class="apple-converted-space"> </span>that
relates to past events as well as the memory, discovery, collection,
organization, presentation, and interpretation of information about these
events. Scholars who write about history are called<span class="apple-converted-space"> </span><span style="color: black; mso-themecolor: text1; text-decoration: none; text-underline: none;">historians</span>.
Events occurring prior to written record are considered<span class="apple-converted-space"> </span><span style="color: black; mso-themecolor: text1; text-decoration: none; text-underline: none;">prehistory</span>.<o:p></o:p></span></div>
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<span style="font-family: Arial, sans-serif; font-size: 10.5pt;">History can also refer to the academic discipline which
uses a<span class="apple-converted-space"> </span><span style="color: black; mso-themecolor: text1; text-decoration: none; text-underline: none;">narrative</span><span class="apple-converted-space"> </span>to examine and analyse a sequence of
past events, and objectively determine the patterns of cause and effect that
determine them.</span><sup id="cite_ref-evans1_5-0" style="unicode-bidi: -webkit-isolate;"><span style="font-family: Arial, sans-serif; font-size: 8.5pt;"><a href="http://en.wikipedia.org/wiki/History#cite_note-evans1-5"><span style="color: black; mso-themecolor: text1; text-decoration: none; text-underline: none;">[5]</span></a><a href="http://en.wikipedia.org/wiki/History#cite_note-6"><span style="color: black; mso-themecolor: text1; text-decoration: none; text-underline: none;">[6]</span></a></span></sup><span class="apple-converted-space"><span style="font-family: Arial, sans-serif; font-size: 10.5pt;"> </span></span><span style="font-family: Arial, sans-serif; font-size: 10.5pt;">Historians
sometimes debate the<span class="apple-converted-space"> </span><span style="color: black; mso-themecolor: text1; text-decoration: none; text-underline: none;">nature of history</span><span class="apple-converted-space"> </span>and
its usefulness by discussing the study of the discipline as an end in itself
and as a way of providing "perspective" on the problems of the
present.</span><sup id="cite_ref-evans1_5-1" style="unicode-bidi: -webkit-isolate;"><span style="font-family: Arial, sans-serif; font-size: 8.5pt;"><a href="http://en.wikipedia.org/wiki/History#cite_note-evans1-5"><span style="color: black; mso-themecolor: text1; text-decoration: none; text-underline: none;">[5]</span></a><a href="http://en.wikipedia.org/wiki/History#cite_note-Tosh1-7"><span style="color: black; mso-themecolor: text1; text-decoration: none; text-underline: none;">[7]</span></a><a href="http://en.wikipedia.org/wiki/History#cite_note-8"><span style="color: black; mso-themecolor: text1; text-decoration: none; text-underline: none;">[8]</span></a><a href="http://en.wikipedia.org/wiki/History#cite_note-9"><span style="color: black; mso-themecolor: text1; text-decoration: none; text-underline: none;">[9]</span></a></span></sup><span style="font-family: Arial, sans-serif; font-size: 10.5pt;"><o:p></o:p></span></div>
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<span style="font-family: Arial, sans-serif; font-size: 10.5pt;">Stories common to a particular culture, but not supported
by external sources (such as the tales surrounding<span class="apple-converted-space"> </span><span style="color: black; mso-themecolor: text1; text-decoration: none; text-underline: none;">King Arthur</span>) are usually classified as<span class="apple-converted-space"> </span><span style="color: black; mso-themecolor: text1; text-decoration: none; text-underline: none;">cultural heritage</span><span class="apple-converted-space"> </span>or<span class="apple-converted-space"> </span><span style="color: black; mso-themecolor: text1; text-decoration: none; text-underline: none;">legends</span>,
because they do not support the "disinterested investigation"
required of the discipline of history.</span><sup id="cite_ref-10" style="unicode-bidi: -webkit-isolate;"><span style="font-family: Arial, sans-serif; font-size: 8.5pt;"><a href="http://en.wikipedia.org/wiki/History#cite_note-10"><span style="color: black; mso-themecolor: text1; text-decoration: none; text-underline: none;">[10]</span></a><a href="http://en.wikipedia.org/wiki/History#cite_note-Low1-11"><span style="color: black; mso-themecolor: text1; text-decoration: none; text-underline: none;">[11]</span></a></span></sup><span class="apple-converted-space"><span style="font-family: Arial, sans-serif; font-size: 10.5pt;"> </span></span><span style="font-family: Arial, sans-serif; font-size: 10.5pt;"><span style="color: black; mso-themecolor: text1; text-decoration: none; text-underline: none;">Herodotus</span>,
a 5th-century BC<span class="apple-converted-space"> </span><span style="color: black; mso-themecolor: text1; text-decoration: none; text-underline: none;">Greek historian</span><span class="apple-converted-space"> </span>is considered within the Western
tradition to be the "father of history", and, along with his
contemporary<span class="apple-converted-space"> </span><span style="color: black; mso-themecolor: text1; text-decoration: none; text-underline: none;">Thucydides</span>, helped form the foundations for the modern study
of human history. Their work continues to be read today and the divide between
the culture-focused Herodotus and the military-focused Thucydides remains a
point of contention or approach in modern historical writing. In the Eastern
tradition, a state chronicle the<span class="apple-converted-space"> </span><span style="color: black; mso-themecolor: text1; text-decoration: none; text-underline: none;">Spring and Autumn Annals</span><span class="apple-converted-space"> </span>was known to be compiled from as early
as 722 BC although only 2nd century BC texts survived.<o:p></o:p></span></div>
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<span style="font-family: Arial, sans-serif; font-size: 10.5pt;">Ancient influences have helped spawn variant
interpretations of the nature of history which have evolved over the centuries
and continue to change today. The modern study of history is wide-ranging, and
includes the study of specific regions and the study of certain topical or
thematical elements of historical investigation. Often history is taught as
part of primary and secondary education, and the academic study of history is a<span class="apple-converted-space"> </span><span style="color: black; mso-themecolor: text1; text-decoration: none; text-underline: none;">major discipline</span><span class="apple-converted-space"> </span>in University studies.<o:p></o:p></span></div>
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History of Humanities

HUMANITIES are academic disciplines that study human culture. The humanities use methods that are primarily critical , or spe...

Operation of Sets

<h1 align="center" style="color: #003399; font-family: Verdana, Arial, Tahoma, sans-serif; font-size: 23px; font-weight: normal; line-height: 30px; margin: 9px 0px 20px;">
Introduction to Sets</h1>
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<div style="margin-bottom: 8px; margin-top: 10px;">
Forget everything you know about numbers.</div>
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In fact, forget you even know what a number is.</div>
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This is where mathematics starts.</div>
<div style="margin-bottom: 8px; margin-top: 10px;">
Instead of math with numbers, we will now think about math with "things".</div>
</div>
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Definition</h2>
<div style="font-family: Verdana, Arial, Tahoma, sans-serif; font-size: 15px; line-height: 23px; margin-bottom: 25px; margin-top: 16px;">
What is a set? Well, simply put, it's <b>a collection</b>.</div>
<div style="font-family: Verdana, Arial, Tahoma, sans-serif; font-size: 15px; line-height: 23px; margin-bottom: 25px; margin-top: 16px;">
First we specify a common property among "things" (this word will be defined later) and then we gather up all the "things" that have this common property.</div>
<div style="float: left; font-family: Verdana, Arial, Tahoma, sans-serif; font-size: 15px; line-height: 23px; margin-bottom: 5px; margin-right: 10px;">
<img height="160" src="http://www.mathsisfun.com/sets/images/set-clothes.jpg" width="200" /></div>
<div style="font-family: Verdana, Arial, Tahoma, sans-serif; font-size: 15px; line-height: 23px; margin-bottom: 25px; margin-top: 16px;">
For example, the items you wear: shoes, socks, hat, shirt, pants, and so on.</div>
<div style="font-family: Verdana, Arial, Tahoma, sans-serif; font-size: 15px; line-height: 23px; margin-bottom: 25px; margin-top: 16px;">
I'm sure you could come up with at least a hundred.</div>
<div style="font-family: Verdana, Arial, Tahoma, sans-serif; font-size: 15px; line-height: 23px; margin-bottom: 25px; margin-top: 16px;">
This is known as a <b>set</b>.</div>
<table border="0" style="color: black; font-family: Verdana, Arial, Tahoma, sans-serif; font-size: 15px; line-height: 23px; width: 100%px;"><tbody>
<tr><td style="line-height: 22px;"><div align="right" style="line-height: 23px; margin-bottom: 25px; margin-top: 16px;">
Or another example is <b>types of fingers</b>.</div>
<div align="right" style="line-height: 23px; margin-bottom: 25px; margin-top: 16px;">
This set includes index, middle, ring, and pinky.</div>
</td><td style="line-height: 22px;"></td><td style="line-height: 22px;"><img height="96" src="http://www.mathsisfun.com/sets/images/set-fingers.jpg" width="120" /></td></tr>
</tbody></table>
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So it is just things grouped together with a certain property in common.</div>
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Notation</h2>
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There is a fairly simple notation for sets. We simply list each element, separated by a comma, and then put some curly brackets around the whole thing.</div>
<div align="center" style="font-family: Verdana, Arial, Tahoma, sans-serif; font-size: 15px; line-height: 23px; margin-bottom: 25px; margin-top: 16px;">
<img alt="Set" height="127" src="http://www.mathsisfun.com/sets/images/set.gif" width="381" /></div>
<div class="def" style="background-color: honeydew; border-bottom-left-radius: 10px; border-bottom-right-radius: 10px; border-top-left-radius: 10px; border-top-right-radius: 10px; border: 1px solid rgb(255, 192, 112); font-family: Verdana, Arial, Tahoma, sans-serif; font-size: 15px; line-height: 23px; margin: 0px 40px; padding: 0px 18px;">
<div align="center" style="margin-bottom: 25px; margin-top: 16px;">
The curly brackets <span class="large" style="color: #a06000; font-size: 19px;">{ }</span> are sometimes called "set brackets" or "braces".</div>
</div>
<div style="font-family: Verdana, Arial, Tahoma, sans-serif; font-size: 15px; line-height: 23px; margin-bottom: 25px; margin-top: 16px;">
This is the notation for the two previous examples:</div>
<div class="indent50px" style="font-family: Verdana, Arial, Tahoma, sans-serif; font-size: 15px; line-height: 23px; margin-bottom: 25px; margin-left: 50px; margin-top: 16px;">
<b>{socks, shoes, watches, shirts, ...}</b><br />
<b>{index, middle, ring, pinky}</b></div>
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Notice how the first example has the "..." (three dots together).</div>
<div class="def" style="background-color: honeydew; border-bottom-left-radius: 10px; border-bottom-right-radius: 10px; border-top-left-radius: 10px; border-top-right-radius: 10px; border: 1px solid rgb(255, 192, 112); font-family: Verdana, Arial, Tahoma, sans-serif; font-size: 15px; line-height: 23px; margin: 0px 40px; padding: 0px 18px;">
<div align="center" style="margin-bottom: 25px; margin-top: 16px;">
The three dots <span class="huge" style="font-size: 35px; font-weight: bold;">...</span> are called an ellipsis, and mean "continue on".</div>
</div>
<div style="font-family: Verdana, Arial, Tahoma, sans-serif; font-size: 15px; line-height: 23px; margin-bottom: 25px; margin-top: 16px;">
So that means the first example continues on ... for infinity.</div>
<div style="font-family: Verdana, Arial, Tahoma, sans-serif; font-size: 15px; line-height: 23px; margin-bottom: 25px; margin-top: 16px;">
<i>(OK, there isn't <b>really</b> an infinite amount of things you could wear, but I'm not entirely sure about that! After an hour of thinking of different things, I'm still not sure. So let's just say it is infinite for this example.)</i></div>
<div style="font-family: Verdana, Arial, Tahoma, sans-serif; font-size: 15px; line-height: 23px; margin-bottom: 25px; margin-top: 16px;">
So:</div>
<ul style="display: table; font-family: Verdana, Arial, Tahoma, sans-serif; font-size: 15px; line-height: 23px; list-style-position: outside; margin: -5px 0px 0px 20px;">
<li style="color: #000088; line-height: 24px; list-style-type: disc; padding: 0px 0px 5px;">The first set <b>{socks, shoes, watches, shirts, ...}</b> we call an <b>infinite set</b>,</li>
<li style="color: #000088; line-height: 24px; list-style-type: disc; padding: 0px 0px 5px;">the second set <b>{index, middle, ring, pinky}</b> we call a <b>finite set</b>.</li>
</ul>
<div style="font-family: Verdana, Arial, Tahoma, sans-serif; font-size: 15px; line-height: 23px; margin-bottom: 25px; margin-top: 16px;">
But sometimes the "..." can be used in the middle to save writing long lists:</div>
<div class="example" style="background-color: #e0f0ff; border-bottom-left-radius: 10px; border-bottom-right-radius: 10px; border-top-left-radius: 10px; border-top-right-radius: 10px; box-shadow: rgb(208, 208, 255) 5px 5px 1px; font-family: Verdana, Arial, Tahoma, sans-serif; font-size: 15px; line-height: 23px; margin: 20px 0px 30px 50px; overflow: hidden; padding: 5px 10px 1px 15px;">
<div style="margin-bottom: 25px; margin-top: 16px;">
Example: the set of letters:</div>
<div align="center" style="margin-bottom: 25px; margin-top: 16px;">
<b>{a, b, c, ..., x, y, z}</b></div>
<div align="center" style="margin-bottom: 25px; margin-top: 16px;">
In this case it is a <b>finite set</b> (there are only 26 letters, right?)</div>
</div>
<h2 style="color: #993300; font-family: qarmic, Verdana, sans-serif; font-size: 20px; font-weight: normal; line-height: 23px; margin: 60px 0px 12px;">
Numerical Sets</h2>
<div style="font-family: Verdana, Arial, Tahoma, sans-serif; font-size: 15px; line-height: 23px; margin-bottom: 25px; margin-top: 16px;">
So what does this have to do with mathematics? When we define a set, all we have to specify is a common characteristic. Who says we can't do so with numbers?</div>
<div class="indent50px" style="font-family: Verdana, Arial, Tahoma, sans-serif; font-size: 15px; line-height: 23px; margin-left: 50px;">
Set of even numbers: {..., -4, -2, 0, 2, 4, ...}<br />
Set of odd numbers: {..., -3, -1, 1, 3, ...}<br />
Set of prime numbers: {2, 3, 5, 7, 11, 13, 17, ...}<br />
Positive multiples of 3 that are less than 10: {3, 6, 9}</div>
<div style="font-family: Verdana, Arial, Tahoma, sans-serif; font-size: 15px; line-height: 23px; margin-bottom: 25px; margin-top: 16px;">
And the list goes on. We can come up with all different types of sets.</div>
<div style="font-family: Verdana, Arial, Tahoma, sans-serif; font-size: 15px; line-height: 23px; margin-bottom: 25px; margin-top: 16px;">
There can also be sets of numbers that have no common property, they are just <b>defined</b> that way. For example:</div>
<div class="indent50px" style="font-family: Verdana, Arial, Tahoma, sans-serif; font-size: 15px; line-height: 23px; margin-left: 50px;">
{2, 3, 6, 828, 3839, 8827}<br />
{4, 5, 6, 10, 21}<br />
{2, 949, 48282, 42882959, 119484203}</div>
<div style="font-family: Verdana, Arial, Tahoma, sans-serif; font-size: 15px; line-height: 23px; margin-bottom: 25px; margin-top: 16px;">
Are all sets that I just randomly banged on my keyboard to produce.</div>
<h2 style="color: #993300; font-family: qarmic, Verdana, sans-serif; font-size: 20px; font-weight: normal; line-height: 23px; margin: 60px 0px 12px;">
Why are Sets Important?</h2>
<div style="font-family: Verdana, Arial, Tahoma, sans-serif; font-size: 15px; line-height: 23px; margin-bottom: 25px; margin-top: 16px;">
Sets are the fundamental property of mathematics. Now as a word of warning, sets, by themselves, seem pretty pointless. But it's only when we apply sets in different situations do they become the powerful building block of mathematics that they are.</div>
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<i>Math can get amazingly complicated quite fast. Graph Theory, Abstract Algebra, Real Analysis, Complex Analysis, Linear Algebra, Number Theory, and the list goes on. But there is one thing that all of these share in common: <b>Sets</b></i>.</div>
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Universal Set</h2>
<table border="0" style="color: black; font-family: Verdana, Arial, Tahoma, sans-serif; font-size: 15px; line-height: 23px;"><tbody>
<tr><td style="line-height: 22px;"><img height="74" src="http://www.mathsisfun.com/sets/images/star.jpg" width="99" /></td><td style="line-height: 22px;"></td><td style="line-height: 22px;">At the start we used the word "things" in quotes. We call this the <b>universal set</b>. It's a set that contains everything. Well, not <i>exactly</i> everything. <b>Everything that is relevant to our question.</b></td></tr>
<tr><td style="line-height: 22px;"><img height="80" src="http://www.mathsisfun.com/sets/images/various-integers.gif" width="100" /></td><td style="line-height: 22px;"></td><td style="line-height: 22px;">Then our sets included integers. The universal set for that would be all the integers. In fact, when doing Number Theory, this is almost always what the universal set is, as Number Theory is simply the study of integers.</td></tr>
<tr><td style="line-height: 22px;"><img height="80" src="http://www.mathsisfun.com/sets/images/various-reals.gif" width="100" /></td><td style="line-height: 22px;"></td><td style="line-height: 22px;">However in Calculus (also known as real analysis), the universal set is almost always the real numbers. And in complex analysis, you guessed it, the universal set is the complex numbers.</td></tr>
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Some More Notation</h2>
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<table align="center" border="0" style="background-color: #cceeff; background-position: initial initial; background-repeat: initial initial; border-collapse: collapse; width: 93%px;"><tbody>
<tr><td style="border: 2px solid white; font-size: 15px; line-height: 22px; padding: 5px;"><img height="80" src="http://www.mathsisfun.com/sets/images/aa.gif" width="60" /></td><td style="border: 2px solid white; font-size: 15px; line-height: 22px; padding: 5px;">When talking about sets, it is fairly standard to use Capital Letters to represent the set, and lowercase letters to represent an element in that set.<br />
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So for example, A is a set, and a is an element in A. Same with B and b, and C and c.</td></tr>
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Now you don't have to listen to the standard, you can use something like <i><b>m</b></i> to represent a set without breaking any mathematical laws (watch out, you can get <b><span style="font-family: Times New Roman, Times, serif;">π</span></b> years in math jail for dividing by 0), but this notation is pretty nice and easy to follow, so why not?</div>
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Also, when we say an element <b>a</b> is in a set <b>A</b>, we use the symbol <img height="12" src="http://www.mathsisfun.com/sets/images/element-symbol.png" width="12" /> to show it.<br />
And if something is not in a set use <img height="14" src="http://www.mathsisfun.com/sets/images/not-element-symbol.png" width="12" />.</div>
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Example: Set <b>A</b> is {1,2,3}. We can see that <b>1 <img height="12" src="http://www.mathsisfun.com/sets/images/element-symbol.png" width="12" /> A</b>, but <b>5 <img height="14" src="http://www.mathsisfun.com/sets/images/not-element-symbol.png" width="12" /> A</b></div>
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Equality</h2>
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Two sets are equal if they have precisely the same members. Now, at first glance they may not seem equal, so we may have to examine them closely!</div>
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Example: Are A and B equal where:</div>
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<li style="color: #000088; line-height: 24px; list-style-type: disc; padding: 0px 0px 5px;">A is the set whose members are the first four positive whole numbers</li>
<li style="color: #000088; line-height: 24px; list-style-type: disc; padding: 0px 0px 5px;">B = {4, 2, 1, 3}</li>
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Let's check. They both contain 1. They both contain 2. And 3, And 4. And we have checked every element of both sets, so: <b>Yes, they are equal!</b></div>
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And the equals sign (=) is used to show equality, so we write:</div>
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A = B</div>
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Subsets</h2>
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When we define a set, if we take pieces of that set, we can form what is called a <b>subset</b>.</div>
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So for example, we have the set {1, 2, 3, 4, 5}. A <b>subset</b> of this is {1, 2, 3}. Another subset is {3, 4} or even another, {1}. However, {1, 6} is not a subset, since it contains an element (6) which is not in the parent set. In general:</div>
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A is a subset of B if and only if every element of A is in B.</div>
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So let's use this definition in some examples.</div>
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Is A a subset of B, where A = {1, 3, 4} and B = {1, 4, 3, 2}?</h3>
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1 is in A, and 1 is in B as well. So far so good.</div>
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3 is in A and 3 is also in B.</div>
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4 is in A, and 4 is in B.</div>
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That's all the elements of A, and every single one is in B, so we're done.</div>
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Yes, A is a subset of B</div>
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Note that 2 is in B, but 2 is not in A. But remember, that doesn't matter, we only look at the elements in A.</div>
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Let's try a harder example.</div>
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Example: Let A be all multiples of 4 and B be all multiples of 2. Is A a subset of B? And is B a subset of A?</h3>
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Well, we can't check every element in these sets, because they have an infinite number of elements. So we need to get an idea of what the elements look like in each, and then compare them.</div>
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The sets are:</div>
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<li style="color: #000088; line-height: 24px; list-style-type: disc; padding: 0px 0px 5px;">A = {..., -8, -4, 0, 4, 8, ...}</li>
<li style="color: #000088; line-height: 24px; list-style-type: disc; padding: 0px 0px 5px;">B = {..., -8, -6, -4, -2, 0, 2, 4, 6, 8, ...}</li>
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By pairing off members of the two sets, we can see that every member of A is also a member of B, but every member of B is not a member of A:</div>
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<br /></div>
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<img alt="pairing off A and B" height="91" src="http://www.mathsisfun.com/sets/images/sets-example2.gif" width="357" /></div>
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So:</div>
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A is a subset of B, but B is not a subset of A</div>
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Proper Subsets</h2>
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If we look at the defintion of subsets and let our mind wander a bit, we come to a weird conclusion.</div>
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Let <b>A</b> be a set. Is every element<b> </b>in <b>A</b> an element in <b>A</b>? (Yes, I wrote that correctly.)</div>
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Well, umm, <i>yes of course</i>, right?</div>
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So doesn't that mean that <b><i>A is a subset of A</i></b>?</div>
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This doesn't seem very <i>proper</i>, does it? We want our subsets to be <i>proper</i>. So we introduce (what else but) <b>proper subsets</b>.</div>
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A is a <b>proper</b> subset of B if and only if every element in A is also in B, and there exists <b>at least one element</b> in B that is <b>not</b> in A.</div>
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This little piece at the end is only there to make sure that A is not a proper subset of itself. Otherwise, a proper subset is exactly the same as a normal subset.</div>
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Example:</h3>
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{1, 2, 3} is a <b>subset</b> of {1, 2, 3}, but is <b>not </b>a<b> proper subset</b> of {1, 2, 3}.</div>
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Example:</h3>
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{1, 2, 3} <b>is </b>a<b> proper subset</b> of {1, 2, 3, 4} because the element 4 is not in the first set.</div>
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Notice that if A is a proper subset of B, then it is also a subset of B.</div>
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Even More Notation</h2>
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When we say that A is a subset of B, we write A <img height="16" src="http://www.mathsisfun.com/sets/images/subset-symbol.png" width="16" /> B.</div>
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Or we can say that A is not a subset of B by A <img height="16" src="http://www.mathsisfun.com/sets/images/not-subset-symbol.png" width="16" /> B ("A is not a subset of B")</div>
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When we talk about proper subsets, we take out the line underneath and so it becomes A <img height="12" src="http://www.mathsisfun.com/sets/images/proper-subset-symbol.png" width="15" /> B or if we want to say the opposite, A <img height="13" src="http://www.mathsisfun.com/sets/images/not-proper-subset-symbol.png" width="16" /> B.</div>
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Empty (or Null) Set</h2>
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This is probably the weirdest thing about sets.</div>
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<img height="180" src="http://www.mathsisfun.com/sets/images/keys-on-guitar.jpg" width="180" /></div>
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As an example, think of the set of piano keys on a guitar.</div>
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<i>"But wait!" you say, "There are no piano keys on a guitar!"</i></div>
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And right you are. It is a set with <b>no elements</b>.</div>
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This is known as the <b>Empty Set</b> (or Null Set).There aren't any elements in it. Not one. Zero.</div>
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It is represented by <img height="13" src="http://www.mathsisfun.com/sets/images/null-set.png" width="16" /></div>
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Or by <b>{}</b> (a set with no elements)</div>
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Some other examples of the empty set are <b>the set of countries south of the south pole</b>.</div>
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So what's so weird about the empty set? Well, that part comes next.</div>
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Empty Set and Subsets</h2>
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So let's go back to our definition of subsets. We have a set A. We won't define it any more than that, it could be any set. <i><b>Is the empty set a subset of A?</b></i></div>
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Going back to our definition of subsets, <b>if every element in the empty set is also in A, then the empty set is a subset of A</b>. But what if we have <b>no</b> elements?</div>
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It takes an introduction to logic to understand this, but this statement is one that is "vacuously" or "trivially" true.</div>
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A good way to think about it is: <i>we can't find any elements in the empty set that aren't in A</i>, so it must be that all elements in the empty set are in A.</div>
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So the answer to the posed question is a resounding <b>yes</b>.</div>
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The empty set is a subset of every set, including the empty set itself.</div>
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Order</h2>
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No, not the order of the elements. In sets it <b>does not matter what order the elements are in</b>.</div>
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Example: {1,2,3,4) is the same set as {3,1,4,2}</div>
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When we say "order" in sets we mean <b>the size of the set</b>.</div>
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Just as there are finite and infinite sets, each has finite and infinite order.</div>
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For finite sets, we represent the order by a number, the number of elements.</div>
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Example, {10, 20, 30, 40} has an order of 4.</div>
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For infinite sets, all we can say is that the order is infinite. Oddly enough, we can say with sets that some infinities are larger than others, but this is a more advanced topic in sets.</div>

Operation of Sets

Introduction to Sets Forget everything you know about numbers. In fact, forget you even know what a number is. This is where mathema...

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