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MA112-Lecture_Notes-OH-L06

# MA112-Lecture_Notes-OH-L06 - MA112 Discrete Mathematics...

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MA112 Discrete Mathematics Dr. Omar Hamdy Summer 2010 1 MA112 Discrete Mathematics Omar Hamdy Lecture Notes 6

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MA112 Discrete Mathematics Dr. Omar Hamdy Summer 2010 2 Review: Sets • Def.: A set is an unordered collection of objects (elements or members). • A set could be represented by: – Enumerating (listing) the set members. – Definition by property (set building notation) • {f(x)| P(x)}, f(x) is a function of x (how the elements look), and P(x) is a propositional function of x.
MA112 Discrete Mathematics Dr. Omar Hamdy Summer 2010 3 Important Sets • Natural Numbers: N = {0,1,2,3,…} • Integers: Z = {…,-3,-2,-1,0,1,2,3,…} • Positive Integers: Z + = {1,2,3,…} • Rational Numbers: Q = {a/b| a,b Z,b 0} • Real Numbers: R = {…,-1,-0.5,0,1, 2,2,…} The Universal Set U • The Empty Set denoted as or { }

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MA112 Discrete Mathematics Dr. Omar Hamdy Summer 2010 4 Sets and Subsets • S is a subset of T if: S T x (x S) (x T) • S is a proper subset of T if: S T S T x ( x S x T ) • For any set S: S – S S
MA112 Discrete Mathematics Dr. Omar Hamdy Summer 2010 5 Set Related Topics Set cardinality Infinite sets Power sets N-tuples Cartesian product Set operations

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MA112 Discrete Mathematics Dr. Omar Hamdy Summer 2010 6 Set Identities • Set Identities are analogous to logical connectives by the following replacements: disjunction ” becomes union conjunction ” becomes intersection negation ” becomes complement “–” • “F” becomes the empty set • “T” becomes the universe of reference U
MA112 Discrete Mathematics Dr. Omar Hamdy Summer 2010 7 Set Identities Identity: – A = A – A U = A Domination – A U = U – A = Idempotent – A A = A – A A = A

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MA112 Discrete Mathematics Dr. Omar Hamdy Summer 2010 8 Set Identities Double Complement: – A = A Commutative – A B = B A – A B = B A Associative – A (B C) = (A B) C – A (B C) = (A B) C
MA112 Discrete Mathematics Dr. Omar Hamdy Summer 2010 9 Set Identities Distributive: – A (B C) = (A B) (A C) – A (B C) = (A B) (A C) DeMorgan – A B = B A – A B = B A Absorption Laws – A (A B) = A – A (A B) = A

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MA112 Discrete Mathematics Dr. Omar Hamdy Summer 2010 10 Set Identities Complement Laws – A A = U – A A =
MA112 Discrete Mathematics Dr. Omar Hamdy Summer 2010 11 Proof Example • Proof that A (B C) = (A B) C • (A B) C = {x | x A B x C } • = {x | (x A x B ) x C } • = {x | x A ( x B x C ) } • = {x | x A (x B C ) } • = A (B C ) • Can you use Venn Diagrams to proof all set identities?

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MA112-Lecture_Notes-OH-L06 - MA112 Discrete Mathematics...

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