Chapter 2:CONCEPT OF SET

7/29/2019 Chapter 2:CONCEPT OF SET

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DISCRETE MATHEMATICS

FORESPEC MEMBER:

Nor Farahana Zainul Hisham (032291)

Nurul Syazwani Kamarulzaman(032691)Nor Shila Latif (032840)

Nor Farahin Rosli (032963)

Nor Irma Fariza Mohd Zamri (033053)

Sam Tau Siong (032261)

Muhammad Husni Ideris (032500)


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CHAPTER 2 :CONCEPT OF SET

Way of listing the elements of Sets

Specifying properties of sets

Set membership

Empty set

Set of numbers (Z,N,etc....)

Set Equality Venn Diagram

Subset

Power Set

Set Operation

Generalised Union and Intersection

Cartesian Product

Second Task: TAF3023


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Way of listing the elements of Sets

1. A written description (Extension)

Example:

A is the set of calendar months beginning with the letter

2. List or Roster Method (Intentional)

Example:

A= { January, June, July}

3. Set builder notation

Example:

A= { x x is a calendar month beginning with the letter J}


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Example:

C is the set of primary colour

C = { Blue, Red, Yellow}

C = {xx is a primary colour}

Blue


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Properties of sets

Sets are inherently unordered: No matter what objects a, b, and c denote,

{a,b,c} ={a,c,b} ={b,a,c} = {b,c,a} = {c,a,b} = {c,b,a}.

All elements are distinct (unequal), multiple listings make nodifference

{a,a,b} = {a,b,b} = {a,b} = {a,a,a,a,b,b,b}.

This set contains at most 2 elements


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Set Membership

Set is an unordered collection of object, the objectinside the set, called elements or members.

We use symbol and to denote the element of

a set.

For example, aA mean a is the element of the set

A. While bA mean b is not the element of set

A.


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Empty Set

There is a special set, that are no element inside it,

which called empty set or null set.

There are 2 symbols to denote an empty set, and{}.

It's quite interested when these 2 symbols combine

together to become {} .It's not a empty set ,it'scalled singleton set. Which represent an empty set in

a set itself.


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Set of numbers

Natural numbers N ={0, 1, 2, 3,...},

Integers Z ={...,2,1, 0, 1, 2,...},

Positive integers Z+ ={1, 2, 3,...},Rational numbers Q ={p/q | p Z,q Z, and q = 0},

real numbers R

Positive real numbers R+,

complex numbers C


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Set Equality

Definition: Set Equality Two sets are equal if they are bothsubsets of one another. The standard notation for equality, =, is

used.

Example:

{1, 2, 3} = {3, 2, 1} = {1, 1, 2, 3, 2, 2}


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Venn Diagram

These are a pictorial representation of sets and a good way to getintuition about (possible)set equalities.

Universal set U= contains all the objects under consideration, is

represented by a rectangle.

Inside this rectangle, circles or other geometrical figures are usedto represent sets.


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Example Venn Diagram

You can also use Venn Diagrams for 3 sets.Let us say the third set is "Volleyball", which drew, glen and jade play:

Volleyball = {drew, glen, jade}

But let's be more "mathematical" and use a Capital Letter for each set:

S means the set of Soccer playersT means the set of Tennis players

V means the set of Volleyball players

The Venn Diagram is now like this:


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Example Venn Diagram

In a school of 320 students, 85 students are in the

band, 200 students are on sports teams, and 60students participate in both activities. How many

students are involved in either band or sports?

25 + 60 + 140 = 225

There are 225 students involved in either band or sports.


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Subset

Encounter situations where the elements of one setare also the elements of a second set.

If all the elements of a set A are also elements of a

set B, then we say that A is a subsetofB, and we

write:

AB

For example:

IfA = {2, 4, 6, 8, 10} and B = {even integers},

thenAB


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Power Set

If we have a set {a,b,c}:

Then a subset of it could be {a} or {b}, or {a,c}, and so

on, and {a,b,c} is also a subset of {a,b,c} .

And the empty set {} is also a subset of {a,b,c}

In fact, if you list all the subsets of S={a,b,c} you willhave the Power Set of {a,b,c}:

P(S) = { {}, {a}, {b}, {c}, {a, b}, {a, c}, {b, c}, {a, b, c} }


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Set OperationUnion

The union of a collection of sets is the set that

contains those elements that are members at least

one set in the collection.

The union of two sets is a new set which combines all

of the members of both sets (and discards duplicates).

Eg: IfA = {1, 2, 3} and B = {4, 5} ,

thenA B = {1, 2, 3, 4, 5} .


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IfA = {1, 2, 3} and B = {4, 5} ,

thenA B = {1, 2, 3, 4, 5} .

1

2

3

4

5


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Set Operation

Intersection

The intersection of a collection of sets is the set

that contains those elements that are members

of all the sets in the collection.

The intersection of two sets is a new set which

only includes those members present in both

sets.

Eg: IfA = {1, 2, 3} and B = {1, 2, 4, 5} ,

thenAB = {1, 2} .


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IfA = {1, 2, 3} and B = {1, 2, 4, 5} ,

thenAB = {1, 2} .

1

23

4

5


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Set Operation

Disjoint Set

Intersection of two sets is the empty set.

Eg: IfA = {1,3,5,7,9} and B = {2,4,6,8,10},

thenAB = , S

1

3

5

79

2

46

8

10


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Set Operation

Set Difference

The difference of set A to B denoted as A-B is setof those elements that are In the set B

i.e: A-B={x:x A and xB}

Similarly B-A={x:x B and xA}

A general A-B B-A

Eg:

If A={a,b,c,d} and B={b,c,d,f} then A-B={a,d} and

B-A={e,f}


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Set Operation

Set Complimentary

The complementary event A is the set of allelements which do not belong to it. It is often

symbolized by A or A. All sampling points of apopulation are either in A or in A'. And no sample

point can be a member of both A and A.


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Set Operation

Characteristics of set

Identity law A U = A

A = ADomination law A U = U

A = Idempotent laws A A = A

A A = A

Complementation law (A) = A

Commutative laws A B = B AA B = B A

Associative laws A (B C) = (A B) CA (B C) = (A B) C

Distributive laws A (B C) = (A B) (A C)A (B C) = (A B) (A C)

De Morgans laws A B = A BA B = A B

Absorption laws A (A B) = AA (A B) = A

Complement laws A A = UA A =


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Generalised Union

Generalised Intersection


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Example Generalised Union and Intersection


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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 by A B is the set of all ordered pairs (a, b) such that a is a

member ofA and b is a member ofB.

Examples:

What is the Cartesian product of A ={1, 2} and B ={a, b, c}?Solution: The Cartesian product A B is

A B ={(1, a), (1, b), (1, c), (2, a), (2, b), (2,c)}.

Note that the Cartesian products A B and B A are not equal, unless A =or B = (so that A B =)or A = B

Some basic properties of Cartesian products:

A = . A (BC) = (AB) (AC). (AB) C= (AC) (BC).

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Chapter 2:CONCEPT OF SET