Set theory and relation

Discrete mathematics

Set Theory And Relation
BY:-
Ankush
kumar

Set Theory
3.1 Sets and Subsets

A well-defined collection of objects
(the set of outstanding people, outstanding is very subjective)
finite sets, infinite sets, cardinality of a set, subset

A={1,3,5,7,9} 1 A,1 B ,1 C
B={x|x is odd}
C={1,3,5,7,9,...}
cardinality of A=5 (|A|=5)
A is a proper subset of B. A B
C is a subset of B. C B

Set Theory
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,b is integer, b not 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

Set Theory

S {A| A is a set and A A}

( a) Show that is S S , then S S.
( b) Show that is S S , then S S

Principia Mathematica by Russel and Whitehead

Set Theory
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]
set equality C D (C D) (D C)
C D (C D D C)
C D D C

Set Theory

null set or empty set : {},
universal set, universe: U

power set of A: the set of all subsets of A

A={1,2}, P(A)={ , {1}, {2}, {1,2}}
If |A|=n, then |P(A)|=2n.

Set Theory

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

For any finite set A with |A|=n 0, there are C(n,k) subsets of
size k.

Counting the subsets of A according to the number, k, of
elements in a subset, we have the combinatorial identity
n n n n
 2n , for n 0
0 1 2 n

Set Theory
common notations
(k) C*=the set of nonzero complex numbers
(l) For any n in Z+, Zn={0,1,2,3,...,n-1}
(m) For real numbers a,b with a<b,
[ a, b] {x R | a x b} closed interval
(a, b) {x R | a x b} open interval
[ a, b ) {x R| a x b}
half-open interval
( a, b ] {x R| a x b}

Set Theory
3.2 Set Operations and the Laws of Set Theory

Def. 3.5 For A,B U
a) A B {x | x A x B} union
b) A B {x | x A x B} intersection
c) A B {x | x A B x A B} symmetric difference
Def.3.6 mutually disjoint A B
Def 3.7 complement A U A {x | x U x A}
Def 3.8 relative complement of A in B
B A {x | x B x A}

Set Theory
Theorem 3.4 For any universe U and any set A,B in U, the
following statements are equivalent:
a) A B
b) A B B reasoning process
c) A B A
(a) (b), (b) (c),
d) B A (c) (d), and (d) (a)

Set Theory
The Laws of Set Theory
(1) A A Law of Double Complement
(2) A B A B Demorgan' s Laws
A B A B
(3) A B B A Commutative Laws
A B B A
(4) A ( B C ) ( A B) C Associativ Laws
e
A (B C) ( A B) C
(5) A ( B C ) ( A B ) ( A C ) Distributi Laws
ve
A (B C) ( A B) ( A C )

Set Theory
The Laws of Set Theory

(6) A A A, A A A Idempotent Laws
(7) A A, A U A Identity Laws
(8) A A U , A A Inverse Laws
(9) A U U , A = DominationLaws
(10) A ( A B ) A Absorption Laws
A (A B) A

Set Theory

s dual of s (sd)

U
U

Theorem 3.5 (The Principle of Duality) Let s denote a theorem
dealing with the equality of two set expressions. Then sd is also
a theorem.

Set Theory

Ex. 3.17 What is the dual of A B?

Since A B A B B . The dual of A B is the dual of
A B B , which is A B B . That is, B A.

Venn diagram A B

U
A A
A B

Set Theory
3.3 Counting and Venn Diagrams

Ex. 3.23. 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 5 A B
|A B | | A| | B | | A B|

20 10 15

Set Theory
3.3 Counting and Venn Diagrams

Ex 3.24. Defect types of an AND gate: B
4 12
D1: first input stuck at 0 11
3 7
43
D2: second input stuck at 0
5
D3: output stuck at 1 A
15
Given 100 samples C
set A: with D1
set B: with D2 | A B C | | A| | B | | C | | A B |
set C: with D3 | A C| | B C| | A B C|
with |A|=23, |B|=26, |C|=30,
| A B | 7, | A C | 8, | B C | 10,
| A B C | 3 , how many samples have defects? Ans:57

Set theory and relation

More Related Content

What's hot (20)

Similar to Set theory and relation (20)

More from ankush_kumar (14)

Recently uploaded (20)

Set theory and relation