- #1

mathmari

Gold Member

MHB

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Let $X$ be an infinite set and let $x\in X$. Show that there exists a bijection $f:X\to X\setminus \{x\}$. Use, if needed, the axiom of choice. To show that $f$ is bijective we have to show that it is surjective and injective.

The axiom of choice is equivalent to saying that, the function $f:X\to X\setminus \{x\}$ is surjective if and only if it has a right inverse. So we have to show that the function $f$ has a right inverse, correct?

Next we have to show that the function is injective.

Is that the way we have to proceed for the proof? Or should we do something else? (Wondering)