09 - Lecture 9: Countability and Uncountability How do we...

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Unformatted text preview: Lecture 9: Countability and Uncountability How do we measure the ‘sizes’ of infinite sets? How can we compare their relative sizes? • The number of elements in a set A is called its car- dinality , denoted | A | . • If | A | is finite, then A is a finite set. Otherwise, A is infinite. • Two sets (finite or infinite) A and B are equinumer- ous if there is a bijection from A to B . (i.e. the elements of A can be paired with the ele- ments of B .) • A set is countably infinite if it is equinumerous with N , the set of natural numbers { , 1 , 2 , ... } . Other- wise, it is uncountably infinite . • A set is countable if it is finite or countably infinite. • The ‘size’ of an uncountable set is much larger than the ’size’ of a countably infinite set. • Examples: N is countably infinite (directly by definition). 2 N is uncountably infinite (to be proved later). 1 Claim: A set A is countably infinite iff its elements can be enumerated as a , a 1 , a 2 , ... Proof: • (if) Suppose the elements of A can be enumerated as a , a 1 , . . . Then, we can define the following function f : N → A , which is a bijection: f ( i ) = a i . Thus, A is countably infinite. • ( only-if ) Suppose A is countably infinite....
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09 - Lecture 9: Countability and Uncountability How do we...

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