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50 views

Help prove $f:X \rightarrow Y$ is an injection $\Leftrightarrow$ $f:X\rightarrow Y$ is a surjection when $|X|=|Y|$

I need to prove: Given non-empty finite sets $X$ and $Y$ with $|X| = |Y|,$ a function $X\rightarrow Y$ is an injection if and only if it is a surjection. The hint given is to use the pigeonhole ...
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2answers
51 views
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1answer
61 views

My first proof employing the pigeonhole principle / dirichlet's box principle - very simple theorem on real numbers. Please mark/grade.

What do you think about my first proof employing the pigeonhole principle? What mark/grade would you give me? Besides, I am curious about whether you like the style. Theorem Among three elements ...
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1answer
61 views

Using The Pigeon-Hole Principle

Let n be a positive integer. Show that in any set of n consecutive integers there is exactly one divisible by n. Here is the solution: Let $a,~a+1,...,a+n-1$ be the integers in the sequence. ...