(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

[tex] f : X \rightarrow Y[/tex] , [tex]g : Y \rightarrow Z[/tex] and [tex]B \subset Z[/tex]

Prove that

[tex]\left(g \circ f\right)^{-1}\left(B \right) = f^{-1} \left({g^{-1} \left(B\right)\right)[/tex].

What is wrong with this proof ?

3. The attempt at a solution

[tex] x_{0} \in \left(g \circ f\right)^{-1}\left(B\right) \Rightarrow g \left(f\left(x_{0} \right)\right)\in B \Rightarrow f \left(x_{0}\right) \in g^{-1} \left(B\right)[/tex]

[tex]f \left(x_{0} \right) \in g^{-1} \left(B\right) \Rightarrow x_{0} \in f^{-1} \left(g^{-1} \left(B \right) \right)[/tex]

Thus,

[tex]\left(g \circ f\right)^{-1}\left(B \right) \subset f^{-1} \left({g^{-1} \left(B \right)\right)[/tex].

I feel uneasy about the proof. I believe my inferences are correct but there is something unsettling about what I did.

Part 2.

Suppose [tex]x_{0} \in f^{-1}\left(g^{-1}\left(B\right)\right)[/tex]

[tex]x_{0}\in f^{-1}\left(g^{-1} \left(B\right)\right) \Rightarrow f \left(x_{0}\right) \in g^{-1}\left(B\right) \Rightarrow g\left(f\left(x_{0}\right)\right) \in B[/tex]

[tex]g \left(f \left(x_{0} \right) \right) \in B \Rightarrow x_{0} \in \left(g \circ f\right)^{-1} \left(B \right)[/tex]

Thus,

[tex]f^{-1} \left({g^{-1} \left(B\right)\right) \subset \left(g \circ f\right)^{-1}\left(B\right)[/tex].

Therefore,

[tex]\left(g\circ f\right)^{-1}\left(B\right) = f^{-1}\left({g^{-1}\left(B\right)\right)[/tex].

Some feedback would be appreciated. I would like to know if this proof is valid and good enough.

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# Homework Help: My sketchy proof on functions and mappings. I need help. ( Latex Fixed!)

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