08B Week 8 - CMPUT 272(Stewart Lecture 14 Reading Epp 6 Set...

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CMPUT 272 (Stewart) Lecture 14 Reading: Epp 6 Set Theory Definition. A set is an unordered collection of distinct elements. S = { 1 , 2 , 5 , 10 , 5 , 4 } set-roster notation 1 S , 5 S , 3 / S element / member / in / not in N = { x Z | x 0 } set-builder notation = {} empty set: the set with no elements Definitions. For sets A and B : A is a subset of B : A B ⇔ ∀ x, (if x A then x B ) A is a proper subset of B : A B A B and x, ( x / A and x B ) A is equal to B : A = B A B and B A Definitions. A finite set is one that has no elements or whose elements can be put into one-to-one correspondence with { 1 , 2 , . . . , n } for some n Z + (see Chapter 7.4). An infinite set is one that is not finite. Definition. The cardinality of a finite set A , denoted | A | , is the number of elements of A . |{ 5 , 1 , 3 , 7 }| = 4 |{ 1 , 2 , 3 , 4 , { 1 , 2 } , { 1 , 2 , 3 } , ∅}| = 7 Definition. The universal set / universe of discourse is the set of all elements in a context. Set operations: Let A and B be subsets of a universal set U . The union of A and B : A B = { x U | x A or x B } The intersection of A and B : A B = { x U | x A and x B } The set difference B minus A : B - A = { x U | x B and x / A } The complement of A : A c = { x U | x / A } Definition. For a set A , the power set of A , denoted P ( A ), is the set of all subsets of A . 1
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Example: Let U = { 1 , 2 , 3 , 4 , 5 , 6 , 7 } be the universal set, W = { 1 , 2 , 3 } , X = { 2 , 3 , 4 } , and Y = { 4 , 5 , 6 } . ∅ ⊆ W the empty set is a subset of every set / W W Y = disjoint sets X c = { 1 , 5 , 6 , 7 } W X Y = { 7 } c | W X Y | = 6 | W X Y | = 0 { 2 , 3 } ⊂ X X Y = { 4 } X - Y = { 2 , 3 } P ( W ) = P ( { 1 , 2 , 3 } ) = { ∅ , { 1 } , { 2 } , { 3 } , { 1 , 2 } , { 1 , 3 } , { 2 , 3 } , { 1 , 2 , 3 } } P ( ) = {∅} |P ( ) | = 1 Example: Let U = { 1 , 2 , 3 , 4 , 5 , 6 , 7 } ∪ P ( { 1 , 2 , 3 , 4 , 5 , 6 , 7 } ) be the universal set, W = { 1 , 2 , 3 } , and Z = {{ 1 , 2 , 3 } , , 1 } .
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