数学 – 集合と写像 - 写像に関する諸概念(合成写像と全射と単射)

学習環境

集合・位相入門(松坂和夫(著)、岩波書店)の第1章(集合と写像)、4(写像に関する諸概念)、問題13.を解いてみる。

問題13.

$\begin{array}{l} y \in B \\ g (y) \in C \\ \exists a_{0} \in A [g ° f (a_{0}) = g (y)] \\ \exists a_{0} \in A [g (f (a_{0})) = g (y)] \\ f (a_{0}) = y \end{array}$
$\begin{array}{l} b_{1}, b_{2} \in B \\ \exists a_{1}, a_{2} \in A [f (a_{1}) = b_{1} \land f (a_{2}) = b_{2}] \\ b_{1} = b_{2} \\ \Rightarrow g (b_{1}) = g (b_{2}) \\ \Rightarrow g (f (a_{1})) = g (f (a_{2})) \\ \Rightarrow g ° f (a_{1}) = g ° f (a_{2}) \\ \Rightarrow a_{1} = a_{2} \\ \Rightarrow f (a_{1}) = f (a_{2}) \\ \Rightarrow b_{1} = b_{2} \end{array}$

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