(a) A nonempty set S is countable if and only if there exists surjective function f:N->S(adsbygoogle = window.adsbygoogle || []).push({});

(b) A nonempty set S is countable if and only if there exists a injective function g:S->N

There are two way proves for both (a) and (b)

(a-1) prove if a nonempty set S is countable, then there exists surjective function f:N->S; (a-2) also prove if there exists surjective function f:N->S, then a nonempty set S is countable

(b-1) prove if a nonempty set S is countable, then there exists a injective function g:S->N; (b-2) also prove if there exists a injective function g:S->N, then a nonempty set S is countable

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A set is countable if it is either finite or denumerable

1) Two finite countable sets are not necessarily of the same cardinality

2) Every two denumerable sets are of the same cardinality.

Set A is denumerable if there is a bijection f:N->A

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How to construct a surjection f:N->S?

Also the inverse of function f which is g:S->N is also injection?

Please help!

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# Prove set S is countable iff there exists a surjective/injective function

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