Prove composition of bijections is bijection

discrete-mathematics functions

Solution 1:

Since they are bijections they have inverses $f^{-1},g^{-1}$. from $B$ to $A$ and from $C$ to $B$

$(g\circ f)\circ(f^{-1}\circ g^{-1})=(g\circ 1_C)\circ g^{-1}=1_C$

$(f^{-1}\circ g^{-1})\circ(g\circ f)=(f^{-1}\circ1_A)\circ f=1_A$

so $g\circ f$ has an inverse and thus is bijective.

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