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# chap02 - 12 Chapter 2 Background material Chapter overview:...

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12 Chapter 2 Background material Chapter overview: Review of set theory Combinatorial methods 2.1 Set theory c ± 2003 Benoˆ ıt Champagne Compiled January 12, 2012

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2.1 Set theory 13 2.1.1 Basic terminology Deﬁnition of a set: A set is a collection of objects (concrete or abstract), called elements, that usually share some common attributes, but are not otherwise re- stricted in any fashion. The curly brackets { and } are used as delimiters when specifying the content of a set. This may be achieved by either listing all the elements of the set explicitly, as is { 1 , 2 , 3 , 4 , 5 , 6 } (2.1) or by stating the common properties satisﬁed by its elements, as in { a : a is a positive integer 6 } (2.2) In the latter case, the notation “ a :” should read “all a such that”. To indicate that an object a is an element of a set A , we write a A ; we also say that a is a member of, or belongs to, A . If a is not an element of A , we write a 6∈ A . Two sets A and B are identical (or equal) if and only if (iﬀ) they have the same elements, in which case we write A = B . If A and B are not identical, we write A 6 = B . Example: Let A = { 1 , 2 ,..., 6 } and B = { 2 , 4 , 6 } . Then A 6 = B because 1 A while 1 6∈ B . c ± 2003 Benoˆ ıt Champagne Compiled January 12, 2012
2.1 Set theory 14 Subset: If every element of a set A is also an element of a set B , we say that A is contained in B , or that A is a subset of B , and write A B . If A is a subset of B but there exists b such that b B and b 6∈ A , we sometimes say that A is a proper subset of B and write A B . The negations of the set relations and are denoted by 6⊆ and 6⊂ , respectively. Example: let A = { 1 , 2 ,..., 6 } , B = { 2 , 4 , 6 } and C = { 0 , 1 } , then B A , B A , C 6⊆ A , etc. Sample space and empty set: In practical applications of set theory, all sets of interest in a given situ- ation are usually subsets of a larger set called sample space, or universal set, and denoted by the letter S . It is also common practice to introduce a degenerate set containing no elements; the latter is called the empty set, or null set, and is denoted by the symbol . c ± 2003 Benoˆ ıt Champagne Compiled January 12, 2012

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2.1 Set theory 15 Theorem 2.1: Let A , B and C denote arbitrary subsets of a sample space S . The following relations hold: (a) A A (b) A B and B C implies A C (c) A = B if and only if A B and B A (d) ∅ ⊆ A S Proof: These basic properties follow directly from the preceding deﬁnitions; their proof is left as an exercise to the reader. ± c ± 2003 Benoˆ ıt Champagne Compiled January 12, 2012
2.1 Set theory 16 Commonly used sets of numbers: Basic sets of numbers: - Positive integers, or natural numbers: N = { 1 , 2 , 3 ,... } - Integers: Z = { 0 , ± 1 , ± 2 } - Rational numbers: Q = { a b : a,b Z and b 6 = 0 } - Real numbers: R - Complex numbers: C = { a + jb : R } , where j = - 1 Note that N Z Q R C .

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## chap02 - 12 Chapter 2 Background material Chapter overview:...

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