Image of a set

Definition

Let $f \colon A \to B$ be a function and let $S$ be a subset of $A$. The image of $S$ under $f$ is the set $$f(S) = \{f(s) \mid s \in S\} = \{b \in B \mid \exists s \in S, \, f(s) = b\}.$$

Remarks

Another notation is $f[S]$. It avoids a confusion with $f(a)$ for an element $a$. The image $f(A)$ of the whole domain is the range of $f$. Example: for $f(x) = x^2$ on $\mathbb{R}$, $f([-1, 2]) = [0, 4]$. The image of a union is the union of the images, but the image of an intersection can be smaller. See image of an intersection.

Explanations

L1 human

To compute $f(S)$, apply $f$ to each element of $S$ and collect the results. For $f(x) = x + 1$ and $S = \{1, 2, 3\}$, the image is $\{2, 3, 4\}$.