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I need to come up with two sets and functions called A, B that satisfy three conditions. The two functions are f: A ⇒ B and g: B ⇒ A. The three conditions are:

(i) Both functions must be onto.

(ii) f(g(x)) = x for all x in B

(iii) There exists y in A such that g(f(y)) ≠ y.

I'm thinking that the two sets should be the set of all positive integers and that only one of the functions should be one-to-one.

AlexK
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1 Answers1

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This can't happen. Let $y\in A$. Since $g$ is onto, we can find $x\in B$ such that $g(x)=y$. Since $f$ is onto we can write $x=f(z)$ for some $z\in A$. Then $$g(f(z))=g(x)=y$$ so that $$f(g(f(z))=f(y)$$ while according to (ii): $$f(g(f(z))=f(z)=x$$ This shows that $x=f(y)$ so that $g(f(y))=g(x)=y$.

Olivier Moschetta
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