ics141-lecture07-Sets

ics141-lecture07-Sets - University of Hawaii ICS141:...

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7-1 ICS 141: Discrete Mathematics I (Spr 2008) University of Hawaii ICS141: Discrete Mathematics for Computer Science I Department of Information and Computer Sciences University of Hawaii Stephen Y. Itoga
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7-2 ICS 141: Discrete Mathematics I (Spr 2008) University of Hawaii Lecture 7 Chapter 2. Basic Structures 2.1 Sets 2.2 Set Operations
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7-3 ICS 141: Discrete Mathematics I (Spr 2008) University of Hawaii Some material in these slides were taken/adapted from the slides made by Prof. Michael P. Frank and Prof. Jonathan L. Gross which are provided through the publisher of “Discrete Mathematics and Its Applications” written by Kenneth H. Rosen. Some slides are from Prof. Baek
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7-4 ICS 141: Discrete Mathematics I (Spr 2008) University of Hawaii Review Review… Variable objects x , y , z ; sets S , T , U . Literal set {a, b, c} and set-builder notation { x | P ( x )}. relational operator, and the empty set . Set relations =, , , , , , etc. Venn diagrams. Cardinality | S | and infinite sets N , Z , R . Next Power sets P( S ). Cartesian product S × T. More set operators: , , - .
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7-5 ICS 141: Discrete Mathematics I (Spr 2008) University of Hawaii The Power Set Operation The power set P( S ) of a set S is the set of all subsets of S . P( S ) :≡ { x | x S }. Examples P({a,b}) = { , {a}, {b}, {a,b}}. S = {0, 1, 2} P(S) = { , {0}, {1}, {2}, {0, 1}, {0, 2}, {1, 2}, {0, 1, 2}} P( ) = { } P({ }) = { , { }} Note that for finite S , |P( S )| = 2 | S | .
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7-6 ICS 141: Discrete Mathematics I (Spr 2008) University of Hawaii Ordered n -tuples These are like sets, except that duplicates matter , and the order makes a difference . For n N , an ordered n-tuple or a sequence or list of length n is written ( a 1 , a 2 , …, a n ). Its first element is a 1 , its second element is a 2 , etc. Note that (1, 2) (2, 1) (2, 1, 1). Empty sequence, singlets, pairs, triples, quadruples, quintuples , …, n -tuples. Contrast with sets’ {}
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7-7 ICS 141: Discrete Mathematics I (Spr 2008) University of Hawaii Cartesian Products of Sets For sets A , B , their Cartesian product denoted by A × B, is the set of all ordered pairs (a, b), where a A and b B. Hence, A × B = {( a , b ) | a A b B } . E.g. {a,b} × {1,2} = {(a,1),(a,2),(b,1),(b,2)} Note that for finite A , B , | A × B |=| A || B | . Note that the Cartesian product is not commutative: i.e. , ¬2200 AB : A × B = B × A .
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This note was uploaded on 09/12/2008 for the course ICS 141 taught by Professor Idk during the Fall '08 term at Hawaii.

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ics141-lecture07-Sets - University of Hawaii ICS141:...

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