Discrete Structure Chapter 1 SetTheory 30

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Chapter 1

Set Theory

1


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Set Theory

1.1 Sets and Subsets

A set is a well-definedcollection of objects

f inite sets, inf ini te sets, cardinali ty of a set, subset

A={1,3,5,7,9}

B={x|xis odd integer} = {x|x integer ganjil}

C={1,3,5,7,9,...}

cardinality ofA=5, no of elements= n(A)=5Ais a proper subset ofB.

Cis a subset ofB.

1 1 1 A B C, ,

A B

C B


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Set Theory


Elements of a setdefined as members of a set.

Capital letters A , B , , Z - to represent sets.

Lowercase letters a , b , c ,, z to represent elements.

Equal settwo sets are equal if and only if they have the same

elements.


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Set Theory


Eg: Elements of a set

A={1,3,5,7,9}

1A , 3 A

2 A


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Set Theory


Def: Subset

If C , D are sets from a universe U , we say that C is a subset of D and we

write C D or D C , if every element of C is an element of D.In addition, D contains an elements that is not in C , then C is called a

proper subset of D , and this is denoted by C D or DC.


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Set Theory


Eg: Let A = { 1 , 2 , 3 , 4 , 5} , C = {1 , 2} , D = {1 , 2}

C is a subset of D - C D or D C

But C is not a proper subset of D -- C D.

C is a proper subset of A --- C A

D is a proper subset of A --- D A

C is equal to D --- C = D


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Set Theory


Eg:Negatex [ x Ax B ]

Soln:x [ x Ax B ]


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Set Theory


If A = B then AB B A

If A B then [AB B A]

[AB B A]

(AB) (B A)

(AB) (B A)


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Set Theory1.1 Sets and Subsets

set equality

C D C D D C ( ) ( )

subsets

A B x x A x B [ ]A B x x A x B

x x A x B

x x A x B

[ ]

[ ( ) )]

[ ]

C D C D D C

C D D C

( )


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Set Theory


null set or empty set : { } ,

universal set, universe: U

power set of A: the set of all subsets ofA

Eg. B={1,2}, thenP(B)={{ }, {1}, {2}, {1,2}}

If n(A)=n, then n(P(A)) = 2n.

Eg. C={1,2,3}, thenP(C)={{ }, {1},{2},{3},{1,2},{1,3},{2,3},{1,2,3}}

Eg. A={1}, thenP(A)={{ }, {1}}


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Set Theory


Eg:Let U = { 1 , 2 , 3 , 4 , 5 , 6 , x , y ,{1,2},{1,2,3},{1,2,3,4}}

And n(U) = 11.

If A = {1,2,3,4} then n(A) = 4. The following statements are TRUE :

a) AU b) {A} U c) AU

d) {A} U e) AU f) {A} U

If B ={5 , 6 , x , y, A} ={5 , 6 , x , y, {1,2,3,4}} then n(B) = 5.

The following statements are TRUE :

a) AB b) {A} B c) {A} B

d) {A}B e) AB f) AB


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Set Theory

1.1 Sets and Subsets common notations

(a) Z=the set of integers={0,1,-1,2,-1,3,-3,...}

(b)N=the set of nonnegative integers or natural numbers

(c) Z+=the set of positive integers(d) Q=the set of rational numbers={a/b| a,bis integer, bnot zero}

(e) Q+=the set of positive rational numbers

(f) Q*=the set of nonzero rational numbers

(g) R=the set of real numbers

(h) R+=the set of positive real numbers(i) R*=the set of nonzero real numbers

(j) C=the set of complex numbers


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Set Theory


Eg: Let U= {1,2,3,4,5,6,7,8,9,10} , A = {1,2,3,4,5} , B = {3,4,5,6,7}, C = {7,8,9}

Find the elements of the following sets:

a) AB b) AB c) B C

d) AC e) A B f) A C

Soln:


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Set Theory

1.2 Set Operations and the Laws of Set Theory

Theorem: Let A , B, C U

a)If AB and B C, then AC

b) If AB and B C, then AC

c) If AB and B C, then AC

d) If AB and B C, then AC


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Set Theory


Theorem:For any universe Uand any setA,Bin U, the

following statements are equivalent:

A B

A B B

A B A

B A

a)

b)

c)

d)

reasoning process

(a) (b), (b) (c),

(c) (d), and (d) (a)


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Set Theory

1.2 Set Operations and the Laws of Set TheoryThe Laws of Set Theory

)()()(

Laws)()()((5)

)()(

Laws)()((4)

Laws(3)

Laws'(2)

ofLaw)1(

CABACBA

veDistributiCABACBA

CBACBA

eAssociativCBACBA

ABBA

eCommutativABBA

BABA

sDemorganBABA

ComplementDoubleAA


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Set Theory


BABA

BABABAAB

BBAABABABA

BABABABABA

BABABABA

BAxBAxxBA

BA

)()()()(

])[(])[()()(

)()()(

)()()()(

}|{

.Negate:Ex


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Set Theory


Def :

i Ii i

i I

i i

A x x A i I

A x x A i I

{ | }

{ | }

for at least one , and

for every

I: index set

Theorem:Generalized DeMorgan's Laws

i I

i

i I

i

i Ii

i Ii

A A

A A


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Set Theory


Eg:For each n+, let An={123,, n-1,n}. Determine

a) An b)An c)An d)Ann=1 n=1 n=1 n=1

7 11 m m

Soln:


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Set Theory


Membership table to establish set equality:

A B AB AB

0

0

11

0

1

01

0

0

01

0

1

11

A A

0

1

1

0


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Set Theory


Eg: Use membership table to establish each of the following

a) AB = AB b) AA = A

c) A(AB) = A d) (AB) (AC) = (A B) (AC)

Soln:


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Set Theory


Eg: Simplify the expression(AB) C B

Soln:


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Set Theory


Eg: Express AB in terms of and

Soln:

A-B = {X|x A and xB} = A B


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Set Theory

1.3 Counting and Venn Diagrams

Ex:In a class of 50 college freshmen, 30 are studying

BASIC, 25 studying PASCAL, and 10 are studying both. How

many freshmen are studying either computer language?

U A B

10 1520

5

| | | | | | | |A B A B A B


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Set Theory


Given 100 samples

setA: with D1

setB: with D2set C: with D3

Ex: Defect types of an AND gate:

D1: first input stuck at 0

D2: second input stuck at 0

D3: output stuck at 1

with |A|=23, |B|=26, |C|=30,

| | , | | , | | ,

| |

A B A C B C

A B C

7 8 10

3 , how many samples have defects?

A

B

C

11 4

35

7

12

15

43

Ans:57

| | | | | | | | | |

| | | | | |

A B C A B C A B

A C B C A B C


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Set Theory


Ex:There are 3 games. In how many ways can one play one game each

day so that one can play each of the three at least once during 5 days?

setA: without playing game 1

setB: without playing game 2

set C: without playing game 3

| | | | | |

| | | | | |

| |

| |

A B C

A B B C C A

A B C

A B C

Ans

2

1

0

3 2 3 1 0 93

3 93 150

5

5

5 5

5

balls containers

1

2

34

5

g1

g2

g3


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End Chapter 1 set theory

31

Discrete Structure Chapter 1 SetTheory 30

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