Unit 6 Sets - Sets Introduction Sets are an important...

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1 Sets – Sets are an important mathematical data structure • Naturally occurring • Theoretically described • Powerful notation – Definition • An unordered collection of distinct objects that share one or more common property Unordered means no inherent order Distinct means there are no duplicate member x[(x S P(x)) (P(x) x S)] Element of Introduction – Sets are denoted with capital letters – Curley brackets are also used {a,b,c} – Empty Set denoted by or {} • not the same as { } – Finite sets can be defined by enumerating the elements of the set – Infinite sets require definition of defining property • { x | P(x) } • Example: { x | x N x 10 } Such that set of all nonnegative integers Notation & Terms
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2 – Other Commonly Used Sets Z : set of all integers Q : set of all rational numbers R : set of all real numbers C : set of all complex numbers Z + : set of all positive integers – Subset (A B) x (x A x B) • A B proper subset x (x A x B) ∧ ∃ x(x B x A) – Equal Sets (A = B) x [(x A x B) (x B x A)] • (A B) (B A) – Power Set ( ρ (S) ) • | (S)| = 2 n where n = |S| • Note: the empty set is a subset of every set – Venn Diagrams – Union (A B) • {x | x A x B} – Intersection (A B) • {x | x A x B} – Complement (A ) • For a set A (S), A is {x | x S x A} – Set Difference (A – B) • { x | x A x B } – Cartesian Product (A × B) • {(x,y) | x A y B} Ordered pair Set Operations
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3 Identity Name A ∪ ∅ = A A U = A Identity Laws A U = U A ∩ ∅ = Domination Laws A A = A A A = A Idempotent Laws (A ) = A Complementation Laws
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This note was uploaded on 04/19/2008 for the course CS 250 taught by Professor Michaelr.wick during the Spring '08 term at Wisc Eau Claire.

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Unit 6 Sets - Sets Introduction Sets are an important...

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