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.
1 Answer
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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 $\endgroup$– user359302Feb 17, 2019 at 13:00
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