Constant mapping

Task number: 2813

Let \(f: X\to X\) be a constant mapping. For which mappings \(g: X\to X\) is it true that \(f \circ g = g\circ f\)?

  • Resolution

    If \(c\) is a value such that \(f(x)=c\) for all \(x\in X\), then \((f\circ g)(x)=f(g(x))=c\) for all \(x\in X\).

    Conversely \((g\circ f)(x)=g(f(x))=g(c)\). If both sides are to be equal, it must be that \(g(c)=c\). The mapping of other values of \(X\) does not matter.

  • Result

    The mapping \(g\) must fulfill \(g(c)=c\), where \(c=Rg(f)\).

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