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Give an example of sets $X, Y$ and functions $f: X \rightarrow Y$ and $g: Y \rightarrow X$ that satisfy the following...

  • $g \circ f$ is a bijection
  • $g \circ f$ is different from the identity function
  • $f$ is not a surjection
  • $g$ is not an injection
  • $X$ contains two elements

Nothing I try satisfies all conditions simultaneously. There must be a better way to solve this than guessing/checking. Any help you can give would be awesome! Thanks!

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I'll start this off, and hopefully you'll see how to keep going.

  • We know $X$ contains 2 elements, so let's just say $X=\{1, 2\}$.

  • Now $g\circ f$ is a bijection - from $X$ to $X$ - which is not the identity. This leaves only one possibility: we must have $g\circ f(1)=2$ and $g\circ f(2)=1$, i.e., $g\circ f$ "flips" $1$ and $2$.

  • OK, now let's start thinking about what we know about $f$ and $g$ from what we've assumed so far. If $g\circ f$ is a bijection, then $f$ must be injective (why?). So our target set $Y$ has at least two distinct elements, $a$ and $b$, with $f(1)=a$ and $f(2)=b$.

  • But $f$ isn't supposed to be surjective. So $Y$ must have at least one more element, say $c$. For simplicity, let's try $Y=\{a, b, c\}$.

  • Now, we're almost done. Do you see how to define a $g$ from $Y$ to $X$ which satisfies all the requirements?

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  • $\begingroup$ So $g(a) = 2$ and $g(b) = 1$ because $f(1) = a$ and $f(2) = b$ and we know, since $g \circ f$ is a bijection and not the identity that $g(f(1)) = 2$ and $g(f(2)) = 1$. Does that work? $\endgroup$
    – Stacy
    Dec 7 '15 at 5:37
  • $\begingroup$ @Stacy Yes, but you haven't yet completely defined $g$ - what should $g(c)$ be? $\endgroup$ Dec 7 '15 at 5:37
  • $\begingroup$ $g$ cannot be an injection, so $g(c)$ must be 1 or 2, but what other piece of evidence limits it to being one or the other? $\endgroup$
    – Stacy
    Dec 7 '15 at 5:41
  • $\begingroup$ @Stacy Nothing whatsoever, either choice is fine. But in answering the question, you do have to make a choice (or at least explicitly say that either choice is fine). $\endgroup$ Dec 7 '15 at 6:07

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