Real Analysis Homework Help

Suppose that n is an element of N and φ: {1,2,..,n} --> {1,2,..,n}.

a) Prove that φ is 1-1 if and only if φ is onto.

b) Suppose that E is a finite set and f: E --> E. Prove that f is 1-1 on E if and only if f takes E onto E.

I’m thinking that this can be done by induction, but I’ve been staring at this for so long now that I can’t even get the problem started. Any hints would be greatly appreciated.

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Originally Posted by erinthered

Suppose that n is an element of N and φ: {1,2,..,n} --> {1,2,..,n}.

a) Prove that φ is 1-1 if and only if φ is onto.

b) Suppose that E is a finite set and f: E --> E. Prove that f is 1-1 on E if and only if f takes E onto E.

I’m thinking that this can be done by induction, but I’ve been staring at this for so long now that I can’t even get the problem started. Any hints would be greatly appreciated.

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Doing homework in the last minute, nice and cool!

Okay, basically there is a property about finite sets.
That if you take a surjection it implies a injection and a injection implies a bijection when you map a set into it set.

Thus, basically all you need to determine whether something is a permutation in a finite set is when it is injective or bijective. It is a useful tool to know.