The document discusses set operations including union, intersection, difference, complement, and disjoint sets. It provides formal definitions and examples for each operation. Properties of the various operations are listed, such as the commutative, associative, identity, and domination laws. Methods for proving set identities are also described.

1. D i s c re te
S tru c tu re s
(Discrete Mathematics)
Topic: Set Operations
©bilalAmjad
bilalamjad78633@yahoo.com

2. Set operations: Union
 Formal definition for the union of two sets:
A U B = { x | x ∈ A or x ∈ B } or
A U B = { x ∈ U| x ∈ A or x ∈ B }
 Further examples
 {1, 2, 3} ∪ {3, 4, 5} = {1, 2, 3, 4, 5}
 {a, b} ∪ {3, 4} = {a, b, 3, 4}
 {1, 2} ∪ ∅ = {1, 2}
 Properties of the union operation
 A∪∅=A Identity law
 A∪U=U Domination law
 A∪A=A Idempotent law
 A∪B=B∪A Commutative law
 A ∪ (B ∪ C) = (A ∪ B) ∪ C Associative law

3. 05/26/12

4. Set operations: Intersection
 Formal definition for the intersection of two sets:
A ∩ B = { x | x ∈ A and x ∈ B }
 Examples
 {1, 2, 3} ∩ {3, 4, 5} = {3}
 {a, b} ∩ {3, 4} = ∅
 {1, 2} ∩ ∅ = ∅
 Properties of the intersection operation
 A∩U=A Identity law
 A∩∅=∅ Domination law
 A∩A=A Idempotent law
 A∩B=B∩A Commutative law
 A ∩ (B ∩ C) = (A ∩ B) ∩ C Associative law

5. Exercise-intersection
05/26/12

6. Exercise-union
05/26/12

7. Disjoint sets
 Formal definition for disjoint sets:
two sets are disjoint if their intersection is the
empty set
 Further examples
 {1, 2, 3} and {3, 4, 5} are not disjoint
 {a, b} and {3, 4} are disjoint
 {1, 2} and ∅ are disjoint
• Their intersection is the empty set
 ∅ and ∅ are disjoint!
• Because their intersection is the empty set

8. Set operations: Difference
 Formal definition for the difference of two sets:
A - B = { x | x ∈ A and x ∉ B }
 Further examples
 {1, 2, 3} - {3, 4, 5} = {1, 2}
 {a, b} - {3, 4} = {a, b}
 {1, 2} - ∅ = {1, 2}
• The difference of any set S with the empty set will be the
set S

9. Complement sets
 Formal definition for the complement of a set:
A = { x | x ∉ A } = Ac
 Or U – A, where U is the universal set
 Further examples (assuming U = Z)
 {1, 2, 3}c = { …, -2, -1, 0, 4, 5, 6, … }
 {a, b}c = Z
 Properties of complement sets
 (Ac)c = A Complementation law
 A ∪ Ac = U Complement law
 A ∩ Ac = ∅ Complement law

10. Set identities
A∪∅ = A A∪U = U
Identity Law Domination law
A∩U = A A∩∅ = ∅
A∪A = A Idempotent Complementation
(Ac)c = A
A∩A = A Law Law
A∪B = B∪A Commutative (A∪B)c = Ac∩Bc
De Morgan’s Law
A∩B = B∩A Law (A∩B)c = Ac∪Bc
A∪(B∪C) A∩(B∪C) =
= (A∪B)∪C Associative (A∩B)∪(A∩C)
Distributive Law
A∩(B∩C) Law A∪(B∩C) =
= (A∩B)∩C (A∪B)∩(A∪C)
A∪(A∩B) =
A Absorption A ∪ Ac = U
Complement Law
A∩(A∪B) = Law A ∩ Ac = ∅
A

11. How to prove a set identity
 For example: A∩B=B-(B-A)
 Four methods:
 Use the basic set identities
 Use membership tables
 Prove each set is a subset of each other
 Use set builder notation and logical equivalences

12. What we are going to prove…
A∩B=B-(B-A)
A B
B-(B-A)
B-A
A∩B

13. Proof by Set Identities
 A ∩ B = A - (A - B) = B – (B – A)
Proof: A - (A - B) = A - (A ∩ Bc)
= A ∩ (A ∩ Bc)c
= A ∩ (Ac ∪ B)
= (A ∩ Ac) ∪ (A ∩ B)
= ∅ ∪ (A ∩ B)
=A∩B

14. Showing each is a subset of the others
 (A ∩ B)c = Ac ∪ Bc
Proof: Want to prove that
(A ∩ B)c ⊆ Ac ∪ Bc and Ac ∪ Bc ⊆ (A ∩ B)c
(i) x ∈ (A ∩ B)c
⇒ x ∉ (A ∩ B)
⇒ ¬ (x ∈ A ∩ B)
⇒ ¬ (x ∈ A ∧ x ∈ B)
⇒ ¬ (x ∈ A) ∨ ¬ (x ∈ B)
⇒x∉A∨x∉B
⇒ x ∈ Ac ∨ x ∈ B c
⇒ x ∈ Ac ∪ B c
(ii) Similarly we show that Ac ∪ Bc ⊆ (A ∩ B)c

15. Exercise