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- 1. CSC102 - Discrete Structures (Discrete Mathematics) Set Operations
- 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
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- 4. Set presentation 4
- 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
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- 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. 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. 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. 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. 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. Set Identities 12
- 13. Set Identities 13
- 14. Set Identities 14
- 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. First Proof A ∩ B = B - (B - A) A B A∩B B-A B-(B-A) 16
- 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. 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. 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
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