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section08 - Section 8 – Cardinality Comparing Sizes of...

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Unformatted text preview: Section 8 – Cardinality Comparing Sizes of Sets It’s clear that the set f 1 ; 2 g is smaller than the set f 1 ; 2 ; 3 g . But which set is bigger N or R ? It turns out that R is bigger MAT 300 Awtrey MATHEMATICS AND STATISTICS 1 / 19 Section 8 – Cardinality Comparing Sizes of Sets It’s clear that the set f 1 ; 2 g is smaller than the set f 1 ; 2 ; 3 g . But which set is bigger N or R ? It turns out that R is bigger much bigger. MAT 300 Awtrey MATHEMATICS AND STATISTICS 1 / 19 Section 8 – Cardinality Comparing Sizes of Sets It’s clear that the set f 1 ; 2 g is smaller than the set f 1 ; 2 ; 3 g . But which set is bigger N or R ? It turns out that R is bigger much bigger. But we need a definition to talk about sizes of sets. MAT 300 Awtrey MATHEMATICS AND STATISTICS 1 / 19 Section 8 – Cardinality Comparing Sizes of Sets It’s clear that the set f 1 ; 2 g is smaller than the set f 1 ; 2 ; 3 g . But which set is bigger N or R ? It turns out that R is bigger much bigger. But we need a definition to talk about sizes of sets. Definition Two sets S and T are called equinumerous and we write S T if there exists a bijective function from S to T . MAT 300 Awtrey MATHEMATICS AND STATISTICS 1 / 19 Section 8 – Cardinality Comparing Sizes of Sets It’s clear that the set f 1 ; 2 g is smaller than the set f 1 ; 2 ; 3 g . But which set is bigger N or R ? It turns out that R is bigger much bigger. But we need a definition to talk about sizes of sets. Definition Two sets S and T are called equinumerous and we write S T if there exists a bijective function from S to T . The word equinumerous literally means same number, implying that S and T have the same number of elements. MAT 300 Awtrey MATHEMATICS AND STATISTICS 1 / 19 Section 8 – Cardinality Comparing Sizes of Sets It’s clear that the set f 1 ; 2 g is smaller than the set f 1 ; 2 ; 3 g . But which set is bigger N or R ? It turns out that R is bigger much bigger. But we need a definition to talk about sizes of sets. Definition Two sets S and T are called equinumerous and we write S T if there exists a bijective function from S to T . The word equinumerous literally means same number, implying that S and T have the same number of elements. Practice Let F be a family of sets. Show that is an equivalence relation on F . MAT 300 Awtrey MATHEMATICS AND STATISTICS 1 / 19 Section 8 – Cardinality Cardinal Numbers A set S is said to be finite if either S = ; or if there exists n 2 N and a bijection f : f 1 ; 2 ; : : : ; n g ! S . If a set is not finite, it is said to be infinite . MAT 300 Awtrey MATHEMATICS AND STATISTICS 2 / 19 Section 8 – Cardinality Cardinal Numbers A set S is said to be finite if either S = ; or if there exists n 2 N and a bijection f : f 1 ; 2 ; : : : ; n g ! S . If a set is not finite, it is said to be infinite ....
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section08 - Section 8 – Cardinality Comparing Sizes of...

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