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I understand that the domain of a function is the set of all input values, but in some textbooks, the domain is an element of that set of input values. How can a set also be a subset of that set?. This is what I'm confused about.

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Not sure, but maybe this answers your question.


In ordinary set-theory it does not happen that a set (e.g. the domain of a function) is an element of itself.

What I did encounter for a function $f:X\to Y$ is the notation $f(X)$.

This however does not mean that $X$ must be looked in that context at as an input value for the function $f$, but is only a notation for the set: $$\{f(x)\mid x\in X\}\subseteq Y$$

This notation is used not only for $X$ itself but also subsets $A\subseteq X$:$$f(A):=\{f(x)\mid x\in A\}\subseteq Y$$

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  • $\begingroup$ Maybe also mention it generalises to arbitrary subsets of X $\endgroup$
    – user359302
    Feb 17, 2019 at 13:00
  • $\begingroup$ @alkchf I followed your advice. $\endgroup$
    – drhab
    Feb 17, 2019 at 13:07

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