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Suppose that $X$ is finite and let $f\colon X\to Y$ be a map. Show that $f$ is injective if and only if $|f(X)| = |X|$

I wrote the elements of $X$ as $x_1,\dots,x_n$ and know that $|X|$ is therefore $n$, but am unsure of what to do next.

egreg
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Craig
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  • The forward implication is pretty straightforward. Suppose f is injective, then the number of elements in X must be exactly of the elements that is mapped over to in Y (that is, the image of X under f). Hence |f(X)| = |X|.

    Now suppose |f(X)| = |X|. We claim that f must be injective. Suppose otherwise. If f is not injective what happens? Try to reach a contradiction.

     – Soby Nov 10 '16 at 12:00
  • Related: http://math.stackexchange.com/questions/2006586/definition-of-terms-for-functions/ Though not exact duplicate, since that question asks about notation clarification. – Arthur Nov 10 '16 at 12:04

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