week14 - Cardinality Part II Original Notes adopted from...

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Unformatted text preview: Cardinality Part II Original Notes adopted from January 8, 2002 (Week 14) c P. Rosenthal , MAT246Y1, University of Toronto, Department of Mathematics Definition. | S | = | T | if there exists f : S → T (one-to-one and onto) Eg . | Even naturals | = | natural | Definition. If | S | = | N | , then S is countably infinite (or denumerable , or enumerable ) and | S | = ℵ (aleph nought). [Elements of S can be listed] | Q | = ℵ Theorem. Union of countable number of countable sets is countable. [0,1] is uncountable | [0 , 1] | = c , | R | = c Definition. | S | 6 | T | if there exists T ⊂ T such that | S 1 | = | T | . ie) There exists f : S → T such that f is one-to-one (not necessarily onto). Schroeder-Bernstein(Cantor) Theorem If | S | 6 | T | and | T | 6 | S | , then | S | = | T | . Proof : Given f : S 1:1 → T , g : T 1:1 → S , we want h : S → T (one-to-one and onto). For s ∈ S , a first ancestor is a t ∈ T such that g ( t ) = s . (If s / ∈ g ( T ), no first ancestor.) A...
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This note was uploaded on 04/26/2011 for the course MATH 246 taught by Professor Applebaugh during the Spring '10 term at University of Toronto.

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week14 - Cardinality Part II Original Notes adopted from...

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