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Unformatted text preview: Sets & Functions Discrete Math MATH 135 David Oury Sets Exercises Functions Exercises Infinite sets Exercises Sequences, Summations Exercises Sets & Functions Discrete Math MATH 135 David Oury February 7, 2012 Chapter 2, Sections 1, 2, 3, 4, 5 of Discrete Mathematics and its Applications, by Rosen Set theory (2 lectures, sections 1, 2) Functions and infinite sets (3 lectures, sections 3, 4, 5) Sets & Functions Discrete Math MATH 135 David Oury Sets Exercises Functions Exercises Infinite sets Exercises Sequences, Summations Exercises Sets I Definition: set A set is an (unordered) collection of distinct things. List: S = { , 1 , 4 , 9 ,... } Form: S = { n 2 : n ∈ N } Property: S = { n ∈ N : ∃ k ∈ N and n = k 2 } English: “The set S of integers that are perfect squares.” A set contains its “elements” or “members.” we write “ 4 ∈ S ” to say “ 4 is an element of S ” we write “ 2 6∈ S ” to say “ 2 is not an element of S ” Sets & Functions Discrete Math MATH 135 David Oury Sets Exercises Functions Exercises Infinite sets Exercises Sequences, Summations Exercises Sets II Cardinality A set S with n distinct elements is said to be finite and to have cardinality n which we denote as  S  = n . An infinite set is one that is not finite. Special sets ∅ = {} the “empty set.” N = { , 1 , 2 ,... } the “natural numbers.” Z = { ..., 2 , 1 , , 1 , 2 ,... } the “integers.” Q = { p/q : p,q ∈ N } the “rational numbers.” R the “real numbers.” Note that π ∈ R and e ∈ R . Notice that N ⊂ Z ⊂ Q ⊂ R . Sets & Functions Discrete Math MATH 135 David Oury Sets Exercises Functions Exercises Infinite sets Exercises Sequences, Summations Exercises Sets III Subsets A set C is a subset of a set A and we write C ⊆ A iff ∀ x ( x ∈ C ⇒ x ∈ A ) . Let A = { a,b,c } . Then ∅ ⊆ A { a,b } ⊆ A A ⊆ A Sets & Functions Discrete Math MATH 135 David Oury Sets Exercises Functions Exercises Infinite sets Exercises Sequences, Summations Exercises Sets IV Equality Sets are equal if and only if they contain the same elements. 1 Order and multiplicity of elements doesn’t matter. For example { 1 , 2 , 3 } = { 2 , 1 , 1 , 3 } . 2 To show that sets A and B are equal we show that A ⊆ B and B ⊆ A. In which case we write A = B . Proper subsets We say “ C is a proper subset of A ” and write C A or C ⊂ A if and only if C ⊆ A and C 6 = A . Sets & Functions Discrete Math MATH 135 David Oury Sets Exercises Functions Exercises Infinite sets Exercises Sequences, Summations Exercises Sets V Power set The power set of a set A is the set P ( A ) of subsets of A . For example the power set of A = { a,b,c } is P ( A ) = {∅ , { a } , { b } , { c } , { a,b } , { a,c } , { b,c } , { a,b,c }} ....
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 Spring '12
 Dr.Oury
 Math, Sets, Natural number, Finite set, David Oury

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