short coursework for

7/31/2019 Short Coursework For

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SHORT COURSEWORK FOR

MATHEMATICS

SET & TRANFORMATION

NAME : NAVAMANI SIMADIRI (891130235230)

PREMASELVI VEERAIAH (900515086506)

UNIT : PEMULIHAN (BM)/MT/BI

COURSE : MT3313 P4 MATHEMATICS 111

LECTURER NAME : ENCIK NORIZAM MOHD. NOR

DATE OF SUBMISSION : 15 SEPTEMBER 2009


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CONTENT

NO CONTENT PAGE

1 QUESTION PAPER

2 COLOBORATION FORM [BORANG MAKLUM

BALAS]

3 CONTENT

4 APPRECIATION

5 HISTROY OF SETS

6 MEANING OF TERM USE IN SET

7 MEANING OF OPERATION

8 FLOW CHART IN SET OPERATION

9 USES OF SETS IN DAILY LIFE

10 USES OF TRANSFORMATION IN DAILY LIFE

11 REFLECTIONS

12 BIBLIOGRAPHY

13 APPENDIX


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APPRECIATION

We are student from unit Pemulihan BT semester 3 had

successfully finished our coursework. Our topic of assignment is set and

transformation.

In this moment, we would like to thank to our lecturer Encik Norizam

Mohd. Nor who really guide us to do this assignment. He gave us a lot of

information about this coursework when we confused. His guide ness really helps

us in the processes finishing this work.


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Beside that, we also like to thank to our parents who really take care

of us when doing the assignment in the holiday. Even though they are really busy

with the work but take initiative of us and help us in complete the course work.

After that, we also would like to thank our college library which really

is the best reference to our course work. Moreover, the book we borrowed also

makes us to get a clear picture on the theory of sets and transformation. They

are a lot of book which we use as good reference. So we would like to thank

again to the IPGM, Raja Melewar Campus library which the good source of

information to this assignment.

On the other hand, we would like to thank our classmates who help

in gather information. By using internet they really cooperative and understanding

each other.

Thank you.

HISTORY OF SET

Mathematical topics typically emerge and evolve through interactions

among many researchers. Set theory, however, was founded by a single paper in

1874 by Georg Cantor: "On a Characteristic Property of All Real Algebraic

Numbers".

Beginning with the work of Zeno around 450 BC, mathematicians had

been struggling with the concept of infinity. Especially notable is the work of

Bernard Bolzano in the first half of the 19th century. The modern understanding
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of infinity began 1867-71, with Georg Cantor's work on number theory. An 1872

meeting between Cantor and Richard Dedekind influenced Cantor's thinking and

culminated in Cantor's 1874 paper.

Cantor's work initially polarized the mathematicians of his day. While

Weierstrass and Dedekind supported Cantor, Kronecker, now seen as a founder

ofmathematical constructivism, did not. Cantorian set theory eventually became

widespread, due to the utility of Cantorian concepts, such as one-to-one

correspondence among sets, his proof that there are more real numbers than

integers, and the "infinity of infinities" ("Cantor's paradise") the power set

operation gives rise to.

The next wave of excitement in set theory came around 1900, when it was

discovered that Cantorian set theory gave rise to several contradictions, called

antinomies orparadoxes. Russell and Zermelo independently found the simplest

and best known paradox, now called Russell's paradox and involving "the set of

all sets that are not members of themselves." This leads to a contradiction, since

it must be a member of itself and not a member of itself. In 1899 Cantor had

himself posed the question: "what is the cardinal numberof the set of all sets?"

and obtained a related paradox.

The momentum of set theory was such that debate on the paradoxes did

not lead to its abandonment. The work ofZermelo in 1908 and Fraenkel in 1922

resulted in the canonical axiomatic set theory ZFC, which is thought to be free of

paradoxes. The work of analysts such as Lebesgue demonstrated the great
http://en.wikipedia.org/wiki/Richard_Dedekindhttp://en.wikipedia.org/wiki/Karl_Weierstrasshttp://en.wikipedia.org/wiki/Leopold_Kroneckerhttp://en.wikipedia.org/wiki/Mathematical_constructivismhttp://en.wikipedia.org/wiki/One-to-one_correspondencehttp://en.wikipedia.org/wiki/One-to-one_correspondencehttp://en.wikipedia.org/wiki/Real_numberhttp://en.wikipedia.org/wiki/Power_sethttp://en.wikipedia.org/wiki/Paradoxhttp://en.wikipedia.org/wiki/Bertrand_Russellhttp://en.wikipedia.org/wiki/Zermelohttp://en.wikipedia.org/wiki/Russell's_paradoxhttp://en.wikipedia.org/wiki/Cardinal_numberhttp://en.wikipedia.org/wiki/Zermelohttp://en.wikipedia.org/wiki/Abraham_Fraenkelhttp://en.wikipedia.org/wiki/ZFChttp://en.wikipedia.org/wiki/Real_analysishttp://en.wikipedia.org/wiki/Henri_Lebesguehttp://en.wikipedia.org/wiki/Richard_Dedekindhttp://en.wikipedia.org/wiki/Karl_Weierstrasshttp://en.wikipedia.org/wiki/Leopold_Kroneckerhttp://en.wikipedia.org/wiki/Mathematical_constructivismhttp://en.wikipedia.org/wiki/One-to-one_correspondencehttp://en.wikipedia.org/wiki/One-to-one_correspondencehttp://en.wikipedia.org/wiki/Real_numberhttp://en.wikipedia.org/wiki/Power_sethttp://en.wikipedia.org/wiki/Paradoxhttp://en.wikipedia.org/wiki/Bertrand_Russellhttp://en.wikipedia.org/wiki/Zermelohttp://en.wikipedia.org/wiki/Russell's_paradoxhttp://en.wikipedia.org/wiki/Cardinal_numberhttp://en.wikipedia.org/wiki/Zermelohttp://en.wikipedia.org/wiki/Abraham_Fraenkelhttp://en.wikipedia.org/wiki/ZFChttp://en.wikipedia.org/wiki/Real_analysishttp://en.wikipedia.org/wiki/Henri_Lebesgue


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mathematical utility of set theory. Axiomatic set theory has become woven into

the very fabric of mathematics as we know it today.

SET THEORY

Set theory is the branch of mathematics that studies sets, which are

collections of objects. Although any type of object can be collected into a set, set

theory is applied most often to objects that are relevant to mathematics.

The modern study of set theory was initiated by Cantorand Dedekind in

the 1870s. After the discovery of paradoxes in informal set theory, numerous
http://en.wikipedia.org/wiki/Mathematicshttp://en.wikipedia.org/wiki/Set_(mathematics)http://en.wikipedia.org/wiki/Georg_Cantorhttp://en.wikipedia.org/wiki/Dedekindhttp://en.wikipedia.org/wiki/Naive_set_theory#Paradoxeshttp://en.wikipedia.org/wiki/Mathematicshttp://en.wikipedia.org/wiki/Set_(mathematics)http://en.wikipedia.org/wiki/Georg_Cantorhttp://en.wikipedia.org/wiki/Dedekindhttp://en.wikipedia.org/wiki/Naive_set_theory#Paradoxes


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TERMS USE IN

SETS

Union of the sets A and B, denoted , is

the set whose members are members of

at least one ofA orB. The union of {1,2, 3} and {2, 3, 4} is the set {1, 2, 3, 4}.

Intersection of the sets A and

B, denoted , is the set whosemembers are members of both

A and B. The intersection of

{1, 2, 3} and {2, 3, 4} is the

set {2, 3}.

Symmetric difference of sets A and B

is the set whose members aremembers of exactly one ofA and B.

For instance, for the sets {1,2,3} and

{2,3,4}, the symmetric difference set

is {1,4}.

Cartesian product ofA and B, denoted, is the set whose members are all

possible ordered pairs (a,b) where a is

a member ofA and b is a member of

B.

Complement of set A relative to set U, denoted , is

the set of all members ofU that are not members of

A. This terminology is most commonly employedwhen U is a universal set, as in the study of

Venn diagrams. This operation is also called the setdifference ofUand A, denoted The complement of{1,2,3} relative to {2,3,4} is {4}, while, conversely,

the complement of {2,3,4} relative to {1,2,3} is

{1}.

Subset . If every member ofset A is also a member of setB, then A is said to be a subsetof B

axiom systems were proposed in the early twentieth century, of which the

ZermeloFraenkel axioms, with the axiom of choice, are the best-known.

The language of set theory is used in the definitions of nearly all

mathematical objects, such as functions, and concepts of set theory are

integrated throughout the mathematics curriculum. Elementary facts about sets

and set membership can be introduced in primary school, along with Venn

diagrams, to study collections of commonplace physical objects. Elementary

operations such as set union and intersection can be studied in this context.

More advanced concepts such as cardinality are a standard part of the

undergraduate mathematics curriculum.

Set theory, formalized using first-order logic, is the most common

foundational system for mathematics. Beyond its use as a foundational system,

set theory is a branch ofmathematics in its own right, with an active research

community. Contemporary research into set theory includes a diverse collection

of topics, ranging from the structure of the real number line to the study of the

consistency oflarge cardinals.

MEANINGS OF TERM USE IN SET
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SYMBOL AND THE MEANING USE IN SET

Symbolically, we use two common methods to write sets. The roster notation is a

complete or implied listing of all the elements of the set. So and

are examples of roster notation defining sets with 4 and

20 elements respectively. The ellipsis, `` '', is used to mean you fill in the

missing elements in the obvious manner or pattern, as there are too many to

actually list out on paper. The set-builder notation is used when the roster

method is cumbersome or impossible. The set B above could be described by

. The vertical bar, ``|'', is read as ``such that'' so

this notation is read aloud as ``the set ofx such that x is between 2 and 40

(inclusive) andxis even.'' (Sometimes a colon is used instead of |.) In set-builder

notation, whatever comes after the bar describes the rule for determining

whether or not an object is in the set. For the set the


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roster notation would be impossible since there are too many real to actually list

out, explicitly or implicitly.

To discuss and manipulate sets we need a short list of symbols commonly used

in print. We start with five symbols summarized in the following table.

The first symbol, , indicates membership of an object in a particular set. The

negation of this, or nonmember ship is often indicated by `` '' (`xis not in

A''). The subset

relation, , states that every element ofA is also an element ofB. Logically,

this would be: if then The union and intersection operators form

new sets by the following rules. The set is defined to be

while is defined to be .

Finally, the complement of a set consists of those objects that are not in the

given set. This presents a minor problem. If


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MEANING OF OPERATIONS USE IN SET

Definition

At the beginning of his Beitrge zur Begrndung der transfiniten

Mengenlehre, Georg Cantor, the principal creator of set theory, gave the

following definition of a set:

By a "set" we mean any collection M into a whole of definite, distinct

objects m (which are called the "elements" ofM) of our perception [Anschauung]

or of our thought.

The elements of a set, also called its members, can be anything: numbers,

people, letters of the alphabet, other sets, and so on. Sets are conventionally

denoted with capital letters. The statement that sets A and B are equal means

that they have precisely the same members (i.e., every member ofA is also a

member ofB and vice versa).

Unlike a multiset, every element of a set must be unique; no two members

may be identical. All set operations preserve the property that each element of
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the set is unique. The order in which the elements of a set are listed is irrelevant,

unlike a sequence ortuple.

As discussed below, in formal mathematics the definition given above

turned out to be inadequate; instead, the notion of a "set" is taken as an

undefined primitive in axiomatic set theory, and its properties are defined by the

ZermeloFraenkel axioms. The most basic properties are that a set "has"

elements, and that two sets are equal (one and the same) if they have the same

elements.

Describing sets

There are two ways of describing, or specifying the members of, a set.

One way is by intensional definition, using a rule or semantic description. See

this example:

A is the set whose members are the first four positive integers.

B is the set of colors of the French flag.

The second way is by extension, that is, listing each member of the set.

An extensional definition is notated by enclosing the list of members in braces:

C= {4, 2, 1, 3}

D = {blue, white, red}
http://en.wikipedia.org/wiki/Sequencehttp://en.wikipedia.org/wiki/Tuplehttp://en.wikipedia.org/wiki/Axiomatic_set_theoryhttp://en.wikipedia.org/wiki/Zermelo%E2%80%93Fraenkel_axiomshttp://en.wikipedia.org/wiki/Intensional_definitionhttp://en.wikipedia.org/wiki/Semantichttp://en.wikipedia.org/wiki/Integerhttp://en.wikipedia.org/wiki/Flag_of_Francehttp://en.wikipedia.org/wiki/Extension_(semantics)http://en.wikipedia.org/wiki/Extensional_definitionhttp://en.wikipedia.org/wiki/Brackethttp://en.wikipedia.org/wiki/Sequencehttp://en.wikipedia.org/wiki/Tuplehttp://en.wikipedia.org/wiki/Axiomatic_set_theoryhttp://en.wikipedia.org/wiki/Zermelo%E2%80%93Fraenkel_axiomshttp://en.wikipedia.org/wiki/Intensional_definitionhttp://en.wikipedia.org/wiki/Semantichttp://en.wikipedia.org/wiki/Integerhttp://en.wikipedia.org/wiki/Flag_of_Francehttp://en.wikipedia.org/wiki/Extension_(semantics)http://en.wikipedia.org/wiki/Extensional_definitionhttp://en.wikipedia.org/wiki/Bracket


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The order in which the elements of a set are listed in an extensional

definition is irrelevant, as are any repetitions in the list. For example,

{6, 11} = {11, 6} = {11, 11, 6, 11}

are equivalent, because the extensional specification means merely that each of

the elements listed is a member of the set.

For sets with many elements, the enumeration of members can be

abbreviated. For instance, the set of the first thousand positive integers may be

specified extensionally as:

{1, 2, 3, ..., 1000},

where the ellipsis ("...") indicates that the list continues in the obvious way.

Ellipses may also be used where sets have infinitely many members. Thus the

set of positive even numbers can be written as {2, 4, 6, 8, ... }.

The notation with braces may also be used in an intensional specification

of a set. In this usage, the braces have the meaning "the set of all ...". So, E=

{playing card suits} is the set whose four members are , , , and . A more

general form of this is set-builder notation, through which, for instance, the set F

of the twenty smallest integers that are four less than perfect squares can be

denoted:

F= {n2 4 : n is an integer; and 0 n 19}
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In this notation, the colon (":") means "such that", and the description can be

interpreted as "F is the set of all numbers of the form n2 4, such that n is a

whole number in the range from 0 to 19 inclusive." Sometimes the vertical bar

("|") or the semicolon (";") is used instead of the colon.

One often has the choice of specifying a set intensionally or extensionally.

In the examples above, for instance,A = Cand B = D.

Membership:

Element (mathematics)

If something is or is not an element of a particular set then this is

symbolised by and respectively. So, with respect to the sets defined above:

4A and 285F(since 285 = 172 4); but

9 Fand green B.

Cardinality

The cardinality | S | of a set S is "the number of members of S." Forexample, since the French flag has three colors, | B | = 3. In mathematicaltheory, a set {45, 6, 7, 768} has a cardinality value of 4.

There is a set with no members and zero cardinality, which is called the

empty set(or the null set) and is denoted by the symbol . For example, the set

A of all three-sided squares has zero members A and thus A = . Though it
http://en.wikipedia.org/wiki/Colon_(punctuation)http://en.wikipedia.org/wiki/Vertical_barhttp://en.wikipedia.org/wiki/Semicolonhttp://en.wikipedia.org/wiki/Element_(mathematics)http://en.wikipedia.org/wiki/Empty_sethttp://en.wikipedia.org/wiki/Colon_(punctuation)http://en.wikipedia.org/wiki/Vertical_barhttp://en.wikipedia.org/wiki/Semicolonhttp://en.wikipedia.org/wiki/Element_(mathematics)http://en.wikipedia.org/wiki/Empty_set


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(respectively A B), whereas other use them to mean the same as A B

(respectivelyAB).

A is a subset ofB

Example:

The set of all men is a proper subset of the set of all people.

{1, 3} {1, 2, 3, 4}.

{1, 2, 3, 4}

{1, 2, 3, 4}.

The empty set is a subset of every set and every set is a subset of itself:

A.

A A.

An obvious but very handy identity, which can often be used to show that two

seemingly different sets are equal:

A = B if and only ifA B and B A.
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Power set

The power set of a set S can be defined as the set of all subsets ofS. This

includes the subsets formed from all the members of S and the empty set. If a

finite set S has cardinality n then the power set ofS has cardinality 2n. The power

set can be written as P(S).

IfS is an infinite (eithercountable oruncountable) set then the power set

ofS is always uncountable. Moreover, ifS is a set, then there is never a bijection

from S onto P(S). In other words, the power set of S is always strictly "bigger"

than S.

As an example, the power set P({1, 2, 3}) of {1, 2, 3} is equal to the set {{1,

2, 3}, {1, 2}, {1, 3}, {2, 3}, {1}, {2}, {3}, }. The cardinality of the original set is 3,

and the cardinality of the power set is 23, or 8. This relationship is one of the

reasons for the terminology power set. Similarly, its notation is an example of a

general convention providing notations for sets based on their cardinalities.
http://en.wikipedia.org/wiki/Countablehttp://en.wikipedia.org/wiki/Uncountablehttp://en.wikipedia.org/wiki/Bijectionhttp://en.wikipedia.org/wiki/Combinatorics#set_sizes_motivate_a_naming_conventionhttp://en.wikipedia.org/wiki/Countablehttp://en.wikipedia.org/wiki/Uncountablehttp://en.wikipedia.org/wiki/Bijectionhttp://en.wikipedia.org/wiki/Combinatorics#set_sizes_motivate_a_naming_convention


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Special sets

There are some sets which hold great mathematical importance and are

referred to with such regularity that they have acquired special names and

notational conventions to identify them. One of these is the empty set. Many of

these sets are represented using blackboard bold or bold typeface. Special sets

of numbers include:

, denoting the set of all primes.

, denoting the set of all natural numbers. That is to say,

(Sometimes .

, denoting the set of all integers (whether positive, negative or zero). So

.

, denoting the set of all rational numbers (that is, the set of all proper

and improper fractions). So, . For example,

and . All integers are in this set since every integera can

be expressed as the fraction .

, denoting the set of all real numbers. This set includes all rational

numbers, together with all irrational numbers (that is, numbers which

cannot be rewritten as fractions, such as , e, and ).
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, denoting the set of all complex numbers.

, denoting the set of all quaternions.

Each of the above sets of numbers has an infinite number of elements, and

each can be considered to be a proper subset of the sets listed below it. The

primes are used less frequently than the others outside of number theory and

related fields.

Basic operations

Unions

There are ways to construct new sets from existing ones. Two sets can be

"added" together. The union ofA and B, denoted byAB, is the set of all things

which are members of eitherA orB.

The union ofA and B, orAB

Examples:

{1, 2} {red, white} = {1, 2, red, white}.

{1, 2, green} {red, white, green} = {1, 2, red, white, green}.
http://en.wikipedia.org/wiki/Complex_numberhttp://en.wikipedia.org/wiki/Quaternionhttp://en.wikipedia.org/wiki/Number_theoryhttp://en.wikipedia.org/wiki/File:Venn0111.svghttp://en.wikipedia.org/wiki/Complex_numberhttp://en.wikipedia.org/wiki/Quaternionhttp://en.wikipedia.org/wiki/Number_theory


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{1, 2} {1, 2} = {1, 2}.

Some basic properties of unions are:

AB = BA.

A (BC) = (AB)C.

A (AB).

AA =A.

A = A.

ABif and only ifAB = B.

Intersections

A new set can also be constructed by determining which members two

sets have "in common". The intersection ofA and B, denoted byA B, is the set

of all things which are members of bothA and B. IfA B = , thenA and B are

said to be disjoint.

The intersection ofA and B, orA B.
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Examples:

{1, 2} {red, white} = .

{1, 2, green} {red, white, green} = {green}.

{1, 2} {1, 2} = {1, 2}.

Some basic properties of intersections:

A B = B A.

A (B C) = (A B) C.

A B A.

A A =A.

A = .

A Bif and only ifA B =A.

Complements

Two sets can also be "subtracted". The relative complement ofA in B

(also called the set theoretic difference ofB andA), denoted by B \A, (orB A) is

the set of all elements which are members ofB, but not members ofA. Note that

it is valid to "subtract" members of a set that are not in the set, such as removing

the element green from the set {1, 2, 3}; doing so has no effect.
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In certain settings all sets under discussion are considered to be subsets

of a given universal set U. In such cases, U \ A is called the absolute

complementor simply complementofA, and is denoted byA.

The relative complement

ofA in B.

The complement ofA in U.

Examples:

{1, 2} \ {red, white} = {1, 2}.

{1, 2, green} \ {red, white, green} = {1, 2}.

{1, 2} \ {1, 2} = .

{1, 2, 3, 4} \ {1, 3} = {2, 4}.
http://en.wikipedia.org/wiki/Universe_(mathematics)http://en.wikipedia.org/wiki/File:Venn1010.svghttp://en.wikipedia.org/wiki/File:Venn0010.svghttp://en.wikipedia.org/wiki/Universe_(mathematics)


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IfU is the set of integers, E is the set of even integers, and

O is the set of odd integers, then the complement ofEin Uis O, or

equivalently, E = O.

Some basic properties of complements:

AA = U.

A A = .

(A) =A.

A \A = .

U = and = U.

A \ B =A B.

Cartesian product

A new set can be constructed by associating every element of one set

with every element of another set. The Cartesian productof two sets A and B,

denoted byA B is the set of all ordered pairs (a, b) such that a is a member of

A and b is a member ofB.

Examples:

{1, 2} {red, white} = {(1, red), (1, white), (2, red), (2, white)}.

{1, 2, green} {red, white, green} = {(1, red), (1, white), (1, green), (2,

red), (2, white), (2, green), (green, red), (green, white), (green, green)}.
http://en.wikipedia.org/wiki/Ordered_pairshttp://en.wikipedia.org/wiki/Ordered_pairs


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{1, 2} {1, 2} = {(1, 1), (1, 2), (2, 1), (2, 2)}.

FLOW CHART IN SET OPERATIONS

(a). Flow chart in subset, universal set and the complement of a set.

Example:

Question: Determine the relationship between subset, universal set and the

complement of the following sets.

P =

Q =

R =

S =

i. ___________ is the universal set.

ii. ___________ is the subset for __________.

iii. ___________ is the complement set for ____________.

Solution:

STEP 1: Draw a Venn diagram

according

to the P, Q, R, and S sets.


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R

. 11

.13

Q

. 10 .

15

P

.

1

2.

1

4

.

1

4

\

STEP 2: Look carefully at the venn

diagram that you draw.

STEP 3: Recognise relations among the

sets.

STEP 4: Answer the question according

to the relations among the sets.


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(b). Flow chart in intersection and union of sets.

Example:

Question:

The diagram below shows the universal set, set A, set B , and set C.

Find the following.

i. A C

ii. A B C

iii. B ( A C )

The answers for the questions are:

S is the universal set.

P is the subset forQ.

Q is the complement set forR.
http://en.wikibooks.org/wiki/File:VennThreeSetsAllocatingElements02.jpg


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Solution: STEP 1: Look carefully at the venn

diagram given.

STEP 2: Recognise relations among the

sets.

STEP 3: Look at set A and set C and

find the intersection between the sets.

STEP 4: Look at set A, set B and set C

and find the union among the sets.

STEP 5: Repeat the step 3 and find the

union with set B.


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2 USES OF SET IN OUR DAILY LIFE

1. There are so many ways to use Venn Diagrams as a means of

communicating data. But Venn Diagrams can be especially utilized in the

elementary classrooms across the curriculum. Of course, Venn Diagrams have

much more mathematical meaning and purpose than simply serving as a visual

way of explaining material. But to utilize the Venn Diagram to prove deductive

arguments logic is extremely important.

Venn Diagrams are very useful when sorting information. For instance, if

a student wanted to sort how many classmates preferred Pepsi to Sprite, or visa

The answers for the questions are:

A C =

A B C

B ( A C )=


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versa, and who liked both equally, a Venn Diagram would be perfect for showing

those preferences.

Students that chose Sprite would be in the portion of the circle that was

labeled S for Sprite; those choosing Pepsi would be placed in the P side of the

circle. Those that liked both equally would be placed in the part of the two circles

that overlapped each other.

2. Classifying foods into food groups can be challenging for children at young

ages. As students start understanding that a food is in a certain food group, they

are then faced with the fact that a certain food may be in two food groups at one

time. It would be much easier for them to understand if they could visualize it.

That is where using the Venn Diagram to classify the foods is such an

advantage. If a student were having trouble classifying the vegetables and fruits,

this using this Venn Diagram would be most useful. Explain that vegetables

would be marked on the side labeled vegetables, and fruit would be marked in

the side marked fruit. If any foods do go in both categories, then those would

be marked in the part of both circles that overlap.

A more in depth Venn diagram would be categorize into three types of

food. For example: food, vegetables, and fruits. I this

Venn Diagram, the student would be able to


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classify all three food groups, and specify if any fell into two of three of the

groups.

If a certain food fell into all three food groups, then it would be marked in

the very middle of all three circles (the part where they all overlap each other).

Of course, there are many other ways to classify information other than

using the Venn Diagram, but none are as easy to understand and grasp the

concept. Venn Diagrams are a useful tool for any and every classroom.


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2 USES OF TRANFORMATION IN DAILY LIFE

Mirror image

1. In geometry, the mirror image of an object ortwo-dimensional figure is the

virtual image formed by reflection in a plane mirror; it is of the same size as the

original object, yet different, unless the object or figure has reflection symmetry

(also known as a P-symmetry).
http://en.wikipedia.org/wiki/Geometryhttp://en.wikipedia.org/wiki/2D_geometric_modelhttp://en.wikipedia.org/wiki/Virtual_imagehttp://en.wikipedia.org/wiki/Reflection_(mathematics)http://en.wikipedia.org/wiki/Plane_mirrorhttp://en.wikipedia.org/wiki/Reflection_symmetryhttp://en.wikipedia.org/wiki/P-symmetryhttp://en.wikipedia.org/wiki/Geometryhttp://en.wikipedia.org/wiki/2D_geometric_modelhttp://en.wikipedia.org/wiki/Virtual_imagehttp://en.wikipedia.org/wiki/Reflection_(mathematics)http://en.wikipedia.org/wiki/Plane_mirrorhttp://en.wikipedia.org/wiki/Reflection_symmetryhttp://en.wikipedia.org/wiki/P-symmetry


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If a point of an object has coordinates (-x, -y,z) then the image of this point

(as reflected from the mirror in y, z plane) has coordinates (-x, y,z) - so mirror

reflection is a reversal of the coordinate axis perpendicular to the mirror's

surface. Thus, a mirror image does not have reversed right and left (or up and

down), but rather reversed front and back.

Two-dimensional mirror images can be seen in the reflections of mirrors or

other reflecting surfaces, or on a printed surface seen inside out.

The word fire and its mirror image are displayed on the front of this fire

engine

2. A text is sometimes deliberately displayed in mirror image, in order to be

read through a mirror. Emergency vehicles such as ambulances or fire engines

use mirror images in order to be read from a driver's rear-view mirror. Some

movie theaters also use a Rear Window Captioning System to assist individuals

with hearing impairments watching the film.

3. Relations between rotation axis, plane of orbit and axial tilt (for Earth).
http://en.wikipedia.org/wiki/Fire_enginehttp://en.wikipedia.org/wiki/Fire_enginehttp://en.wikipedia.org/wiki/Fire_enginehttp://en.wikipedia.org/wiki/Ambulancehttp://en.wikipedia.org/wiki/Rear-view_mirrorhttp://en.wikipedia.org/wiki/Movie_theaterhttp://en.wikipedia.org/wiki/Rear_Window_Captioning_Systemhttp://en.wikipedia.org/wiki/Hearing_impairmenthttp://en.wikipedia.org/wiki/Orbital_plane_(astronomy)http://en.wikipedia.org/wiki/Axial_tilthttp://en.wikipedia.org/wiki/File:Fire-mirror-image.jpghttp://en.wikipedia.org/wiki/Fire_enginehttp://en.wikipedia.org/wiki/Fire_enginehttp://en.wikipedia.org/wiki/Ambulancehttp://en.wikipedia.org/wiki/Rear-view_mirrorhttp://en.wikipedia.org/wiki/Movie_theaterhttp://en.wikipedia.org/wiki/Rear_Window_Captioning_Systemhttp://en.wikipedia.org/wiki/Hearing_impairmenthttp://en.wikipedia.org/wiki/Orbital_plane_(astronomy)http://en.wikipedia.org/wiki/Axial_tilt


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In astronomy, rotation is a commonly observed phenomenon. Stars,

planets and similar bodies all spin around on their axes (the plural of axis). The

rotation rate of planets in the solar system was first measured by tracking visual

features. Stellar rotation is measured through Doppler shift or by tracking active

surface features.

This rotation induces a centrifugal acceleration in the reference frame of

the Earth which slightly counteracts the effect of gravity the closer one is to the

equator. One effect is that an object weighs slightly less at the equator. Another

is that the Earth is slightly deformed into an oblate spheroid.

Another consequence of the rotation of a planet is the phenomenon of

precession. Like a gyroscope, the overall effect is a slight "wobble" in the

movement of the axis of a planet. Currently the tilt of the Earth's axis to its orbital

plane (obliquity of the ecliptic) is 23.45 degrees, but this angle changes slowly

(over thousands of years).
http://en.wikipedia.org/wiki/Astronomyhttp://en.wikipedia.org/wiki/Starhttp://en.wikipedia.org/wiki/Planethttp://en.wikipedia.org/wiki/Stellar_rotationhttp://en.wikipedia.org/wiki/Doppler_shifthttp://en.wikipedia.org/wiki/Centrifugal_force_(fictitious)http://en.wikipedia.org/wiki/Equatorhttp://en.wikipedia.org/wiki/Oblate_spheroidhttp://en.wikipedia.org/wiki/Precessionhttp://en.wikipedia.org/wiki/Gyroscopehttp://en.wikipedia.org/wiki/Earthhttp://en.wikipedia.org/wiki/Obliquity_of_the_ecliptichttp://en.wikipedia.org/wiki/Astronomyhttp://en.wikipedia.org/wiki/Starhttp://en.wikipedia.org/wiki/Planethttp://en.wikipedia.org/wiki/Stellar_rotationhttp://en.wikipedia.org/wiki/Doppler_shifthttp://en.wikipedia.org/wiki/Centrifugal_force_(fictitious)http://en.wikipedia.org/wiki/Equatorhttp://en.wikipedia.org/wiki/Oblate_spheroidhttp://en.wikipedia.org/wiki/Precessionhttp://en.wikipedia.org/wiki/Gyroscopehttp://en.wikipedia.org/wiki/Earthhttp://en.wikipedia.org/wiki/Obliquity_of_the_ecliptic


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My name is Navamani d/o Simadiri. I am from Pemulihan Bahasa Tamil

unit. I am in 3rd semester this year. First of all, Im very glad to thank Mr. Norizam


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Mohd Nor who is my Mathematics lecturer. He guide and help me a lot in doing

this assignment. Even though, sometimes I confused about the assignment, but

he will try to give more explanation on how to do this assignment easily.

He gave this assignment on 20th August 2009. This assignment must do in

a group of 2 to 5. We are 2 people in a group. So, my group member is

Premaselvi d/o Veeraiah. We work together and successfully finished this

coursework on 13th September 2009. It took about three weeks to complete the

work. We got the assignment on set and transformation. Firstly, my partner and

I feel very confused on how to do the assignment but after read the question for

several time and with our lectures help we got a clear picture on how to do the

coursework.

We start our coursework by searching informations about set and

transformation. My group member and I went to our college library to find books

and materials about the topics. Luckily, we managed to collect a lot of

informations about those two topics. We borrowed some mathematics reference

books which is really helping us in this coursework.

Other than that, we also search a lot of informations about set and

transformation in the internet. The informations that we search in the internet are

history of set, definition of set, uses of set and transformation in daily life. We had

a hard time to search the uses of set and transformation in daily life. We also find

and create some examples to explain the operation of set.


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Besides that, I think that this coursework really help me by increasing my

knowledge and understanding on set and transformation. I get the some useful

informations that I never been thought about such as the history of set and uses

of set and transformations in daily. The informations really interesting and attract

me a lot.

Furthermore, I also realize the cooperation, toleration, and hardworking

among us is very important to finish this coursework at the time. From this kind of

assignment I strongly believe that, my mathematical skills in set and

transformation will be better soon. Its also will help me when I become a teacher.

My name is Premaselvi d/o Veeraiah. I am from Pemulihan Bahasa

Tamil / Mathematics / English unit. I am in semester 3 this year. I received a

maths assignment titled set and transformations. I received this assignment on

20th August 2009. We do this coursework in a group of 2 to 5. We are only two


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people in my group. My group member who worked with me is Navamani d/o

Simadiri.

Firstly, this coursework made me to learn more about set and

transformation. Starting, I felt a bit worry about this coursework, but it clear after

listened explanations from my lecturer. Then, I felt very easy on doing the

assignment. We start our coursework by collecting the details and materials

which related to set and transformation. We choose a day to collect the details.

We went to IPRM library to collect informations about our assignment. We

managed to find a lot of books which related to the both topics that given to us.

We just borrowed the books and refer it every time when to do the assignment. In

addition, we also try to get more quality and accurate informations by surfing the

net. We get a lot of good informations such as history set, uses of set and

transformations in daily life and meanings of set operations. All the informations

were really different.

By doing this coursework, I am now able to solve set and transformations

questions very easily. I can form a lot of strategies to solve a problem involving

set and transformations by doing this coursework. Besides that, I learn to

cooperate with my group members as I face a lot of difficulties to do it in many

ways. There are many details about set and transformations that I have learned.

This will help me later when me teaching the students in school.

So I want to thank my lecturer Mr. Norizam Mohd Nor and my friends for

helping me to do this assignment. I hope to do more assignments like this again


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in future to improve my mathematical skills. I also strongly believe that those

information about the set and transformations sure will help me in my future

when I become a mathematics teacher.

BIBLIOGRAPHY

www.yahoo.com.my

www.google.com.my

www.msn.com.my

http://www.histroy of set. com.my
http://www.yahoo.com.my/http://www.google.com.my/http://www.msn.com.my/http://www.yahoo.com.my/http://www.google.com.my/http://www.msn.com.my/


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http://www.operation of set

http://www.uses of set.com.my

http://www.uses of transformation.com.my

http://www. meaning of operation of set.com.my

Advanced General Mathematics. 1999. Cambrige

Mathematics Form 4. 2006. Penerbitan Mega Setia Emas SDN.BHD.
http://www.operation/http://www.uses/http://www.uses/http://www/http://www.operation/http://www.uses/http://www.uses/http://www/


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short coursework for

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