Inverse of a composition

Task number: 2441

Show that in each group holds \((a\circ b)^{-1}=b^{-1}\circ a^{-1}\).

  • Resolution

    Oserve first that \((b^{-1}\circ a^{-1})\circ(a\circ b)= b^{-1}\circ (a^{-1} \circ a) \circ b= b^{-1}\circ e \circ b= b^{-1}\circ b= e\).

    Hence both \((a\circ b)^{-1}\) and \(b^{-1}\circ a^{-1}\) are inverses of \(a\circ b\). Therefore they must be equal.

Difficulty level: Easy task (using definitions and simple reasoning)
Proving or derivation task
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