Discrete Mathematics and Its Applications with MathZone

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[email protected], modified from [email protected] UCI ICS/Math 6D, Fall 2007 2-Sets+Functions-1 Sets “Set”=Unordered collection of Objects “Set Elements” = “Members of the Set” = Objects in Set X ={2,3,5,7,11,13} 3 X X [“3 is a member of X”] [“3 is a member of X”] 4 X X [“4 is not a member of X”] [“4 is not a member of X”] Common sets of numbers: N, Z, Q, R Sets are “equal” if and only if (iff) they have the same elements A = B [ V x (x A A x x B) B) ] ] A is a “subset ” of B, A B, iff every element in A is also in B: A B [ V x (x A A x x B) B) ] ] Don’t confuse with : x A iff {x} A A is a “proper subset ” of B: (A B) iff A B A B. ={} , the null/empty set: V X ( X), but for many X’s, X tricky: If X ={ } then X and { }
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[email protected], modified from [email protected] UCI ICS/Math 6D, Fall 2007 2-Sets+Functions-2 Ordered n-tuples Sets are unordered collections; with sets, ordering and duplication don’t matter {0,5,3,4,6,2,5,1,3,4,3} ={0,1,2,3,4,5,6} Ordering matters with (ordered) n-tuples (x,y,z,t,v)=(1,4,3,2,4) is short-hand for 5 equalities: x=1, y=4, z=3, t=2, v=4
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[email protected], modified from [email protected] UCI ICS/Math 6D, Fall 2007 2-Sets+Functions-3 Set Builder Notation Building sets from Properties S = {x | P(x)}, where P( ) is a predicate i.e. S is a set of all x’s which s.t. P(x) is True “Explicit universe” version: S = {x U | P(x)}: Set of all x’s in U s.t. P(x) is True. i.e. S = {x U | P(x)} if and only if V x (x U P(x)) x S Examples: {x N | 3x<12} = {0,1,2,3} N | 3x<12} = {0,1,2,3} {x N | N | 5 n N x=2n} = {0,2,4,6,8,...} = N x=2n} = {0,2,4,6,8,...} = Even natural numbers Even natural numbers {x N | N | 5 n N x-1=2n} = {1,3,5,7,9,...} = N x-1=2n} = {1,3,5,7,9,...} = Odd natural numbers Odd natural numbers
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[email protected], modified from [email protected] UCI ICS/Math 6D, Fall 2007 2-Sets+Functions-4 Building Sets from Sets |S| = “Cardinality” of set S = number of (distinct) elements in set S e.g.: |{a,b,c}| = 3, |{a,b}|=2, |{c}|=1, e.g.: |{a,b,c,a,b,a,a,c}| = |{a,b,c}| = 3 e.g.: | | = 0 (because “ ” denotes an empty set!) e.g.: |{ }| = 1 (because “{ }” denotes a set with one element x, which happens to be an empty set, i.e. x= ) e.g.: |{ ,{ }}| = 2 (set of two elements, x and y, where x=
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