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If we have two functions, $f:\mathbb{Z}\rightarrow\mathbb{Z}$ and $g:\mathbb{Z}\rightarrow\mathbb{Z}$ where $g(f(a))=a$ for every integer $a$, how do we satisfy these conditions so that $f$ and $g$ are not bijective?

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    @johnnymath: Martin Sleziak's point was to use the function on $\mathbb{N}$ from Alon Amit's post, then extend it to negatives with the identity. The section on $\mathbb{N}$ has already violated bijectivity.2011-11-18

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First, $f$ will have to be injective. If $f$ is not injective then two integers $n$ and $m$, have $f(n)=f(m)$ and so $g(f(n))=g(f(m))$. Similarly $g$ has to be surjective, or else for some $n$, $g(f(n))\neq n$. But, $f$ can fail to be surjective and $g$ can fail to be injective. Suppose that $f(n)=2n$ and $g(n)=\frac{n}{2}$ if $n$ is even and $g(n)=0$ if $n$ is odd. Then $g(f(n))=n$.