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If f is an injection from $A$ to $B$, then it follows that there is a left inverse, $g$, from $B$ to $A$ where $g(f(a)) = a$. 1) Can I say that f is a right inverse of $g$? 2) How do I justify that?

Hamou
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user3555927
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  • Welcome to math.SE: since you are new, I wanted to let you know a couple things about the site. In order to get the best possible answers, it is helpful if you say in what context you encountered the problem, and what your thoughts on it are. Also, you might be interested to know that you can use Latex commands to format the mathematics in your question. – Jessica B Oct 04 '14 at 06:48
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2 Answers2

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If $f:A\rightarrow B$ and $g:B\rightarrow A$ are functions and $1_A:A\rightarrow A$ is prescribed by $a\mapsto a$ then the following statements are equivalent:

  • $g\circ f=1_A$
  • $g$ is a left-inverse of $f$
  • $f$ is a right-inverse of $g$

If this is the case then $f$ is injective and $g$ is surjective.

drhab
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Yes , by the definition of right inverse , $f$ is a right inverse of $g$ $\iff$ $gof=id$ , with $id$ been the identity

gdlm
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