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lecture11

# lecture11 - Lecture 11 Sets and Functions Recap Set...

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Lecture 11 Sets and Functions

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Recap Set equality: A = B iff x (x A x B) Or, equivalently A = B iff ((A B) (B A)) Subset Relationship S T iff x (x S x T) Cardinality of S: |S| Power set of S: P(S)
Using logic to define set operations A B = {x | (x A) (x B)} A B = {x | (x A) (x B)} Difference: A – B = {x | x A x B} Complement } | { A x x A

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Logic – Set Theory Analogies Union disjunction (OR) Intersection conjunction (AND) Complement negation (NOT) T U (everything) F (nothing)
A = A A U = A Identity Law A U = U A = Domination law A A = A A A = A Idempotent Law (A c ) c = A Double negation Law A B = B A A B = B A Commutative Law (A B) c = A c B c (A B) c = A c B c De Morgan’s Law A (B C) = (A B) C A (B C) = (A B) C Associative Law A (B C) = (A B) (A C) A (B C) = (A B) (A C) Distributive Law A (A B) = A A (A B) = A Absorption Law A A c = U A A c = Complement Law Set Identities

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Proving Set Identities Show that A (B-A) = A B
Maintaining Order A set is an unordered collection of elements If the order matters, we need a different structure The ordered n-tuple is an ordered collection (a 1 , a 2 , …, a n ) 2-tuple = pair (think of coordinates on the plane)

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lecture11 - Lecture 11 Sets and Functions Recap Set...

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