Successfully reported this slideshow.

# 8-Sets-2.ppt

Upcoming SlideShare
Set Operations
×

# 8-Sets-2.ppt

Disc

Disc

## More Related Content

### 8-Sets-2.ppt

1. 1. CSC102 - Discrete Structures (Discrete Mathematics) Set Operations
2. 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 2
3. 3. 3
4. 4. Set presentation 4
5. 5. 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
6. 6. 6
7. 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 7
8. 8. Exercise • What is the cardinality of AUB where A and B are two finite sets? • |A ∪ B| = |A| + |B| ??? • Incorrect • |A ∪ B| = |A| + |B| − |A ∩ B|??? • Correct 8
9. 9. Set operations: Difference Formal definition for the difference of two sets: A - B = { x | x  A and x  B } Sometimes denoted by AB. 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
10. 10. Exercise • What is the cardinality of A∩B where A and B are two finite sets? • |A∩B | = |B| - |B - A|??? • Correct • |A∩B | = |A| - |A - B|??? • Correct • |A∩B | = |A| + |B| - (|A - B| + |B - A|)??? • Incorrect 10
11. 11. 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 11
12. 12. Set Identities 12
13. 13. Set Identities 13
14. 14. Set Identities 14
15. 15. How to prove a set identity For example: A ∩ B = B - (B - A) Four methods: 1. Using the basic set identities 2. Proving that each set is a subset of each other 3. Using set builder notation and logical equivalences 4. Using membership tables (we’ll not study it) • Similar to truth tables 15
16. 16. First Proof A ∩ B = B - (B - A) A B A∩B B-A B-(B-A) 16
17. 17. 1. Proof by Set Identities A  B = A - (A - B) = B – (B – A) Proof: A - (A - B) = A - (A  Bc) A - B = A  Bc = A  (A  Bc)c Same as above = A  (Ac  B) De Morgan’s Law = (A  Ac)  (A  B) Distributive Law =   (A  B) Complement Law = A  B Identity Law 17
18. 18. 2. Showing each is a subset of the others Second Example Proof  (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  Bc  x  Ac  Bc (ii) Similarly we show that Ac  Bc  (A  B)c 18
19. 19. Same proof with Set builder Notation (A ∩ B)c = Ac ∪ Bc • (A ∩ B)c = {x | x  A ∩ B} Definition of complement = {x |￢(x ∈ (A ∩ B))} Definition of does not belong symbol = {x |￢(x ∈ A ∧ x ∈ B)} Definition of intersection = {x | ￢(x ∈ A)∨￢(x ∈ B)} De Morgan’s law = {x | x  A ∨ x  B} Definition of does not belong symbol = {x | x ∈ Ac ∨ x ∈ Bc} Definition of complement = {x | x ∈ Ac ∪ Bc} Definition of union = Ac ∪ Bc meaning of set builder notation 19
20. 20. 20
21. 21. END 21