lecture23 - MA1100 Lecture 23 Cardinality Equivalence of...

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MA1100 Lecture 23 Cardinality Equivalence of sets Infinite sets Denumerable sets Countable sets Uncountable sets Chartrand: 10.1, 10.2, 10.3
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Which set is larger? Question: • They are both infinite sets • Can we compare their “size”? N (0,1) 2 Lecture 23
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Equivalence of sets Recall Let A and B be two sets. Define the relation: A º B if and only if there exists a bijection f: A Ø B from A to B. This is an equivalence relation. p q r a b c AB x y z C Example 3 Lecture 23 reflexive A º A symmetric A º B B º A transitive A º B, B º C A º C
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Equivalent sets and cardinality Proposition Then A is equivalent to B if and only if |A| = |B|. Let A and B be finite sets. ( )I f |A| = |B|, then A is equivalent to B . ( f A is equivalent to B, then |A| = |B| . 4 Lecture 23 If there exists a bijection f: A Ø B from A to B, we say A is (numerically) equivalent to B. Definition
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Equivalent sets and cardinality Definition We define |A| = |B| if A is equivalent to B. Let A and B be infinite sets. there exists a bijection f: A Ø B So |A| |B| A is not equivalent to B. there is no bijection f: A Ø B 5 Lecture 23 A is equivalent to B |A| = |B|
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Positive odd numbers Example Let f: N Ø D + be defined by f(x) = 2x – 1 . Let D + be the set of positive odd numbers . So | N | = |D + |. D + = {1, 3, 5, …} Then f is a bijection . f is an injection : Let x, y œ N . f(x) = f(y) 2x – 1 = 2y – 1 x = y f is a surjection : Let d œ D + . So d is positive odd. Then d = 2n - 1 for some integer n ¥ 1.
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lecture23 - MA1100 Lecture 23 Cardinality Equivalence of...

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