# How do we prove that a mod function is $1-1$? [closed]

in a specified subset of $Z$ (otherwise, it wouldn't be $1-1$)

do we have to show that $f(x) = f(y)$?

i don't see how it can be done since we can manipulate modular expressions algebraically.

x mod b = x mod b

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The general framework is the same: Assume $f(x) = f(y)$ and show $x = y$. –  Austin Mohr Oct 28 '12 at 23:23
Your question is likely to be closed in its current form. Do you have an explicit example you are working on? –  Austin Mohr Oct 28 '12 at 23:31
If by mod you mean absolute value function, then it is NOT one-to-one. If you are referring to modular arithmetic then it is NOT either, that is why it is modular. I think it is obvious that remainders are not unique. For example, $23\bmod 12 = 47\bmod 12 = 11$, but $23 \ne 47$. However, it is onto.