# Can the domain also be referred to as the element of the set?

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.

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 $$A\subseteq X$$:$$f(A):=\{f(x)\mid x\in A\}\subseteq Y$$

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