Question

Proposition 16.4 Let S be a non–empty finite set. (a) There is a unique n 2...

Proposition 16.4 Let S be a non–empty finite set.
(a) There is a unique n 2 N1 such that there is a 1–1 correspondence from {1, 2,...,n} to S.
We write |S| = n. Also, we write |;| = 0.
(b) If B is a set and f : B ! S is a 1–1 correspondence, then B is finite and |B| = |S|.
(c) If T is a proper subset of S, then T is finite and |T| < |S|.

Prove part b.

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Answer #1

we have to prove part (b)

is a one to one correspondence

we know the definition of one to one map if or   

so image of i.e   then is finite

and given set is finite , and there is a one to one correspondence between to finite set ,

hence

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