What does this symbol mean $f(A \cap B)?$

Question

What does $f(A \cap B)?$ mean. Is it a set, function , polynomial, etc? What are the characteristics in its structure? what does this mean in detail?

Answer 1

If $f:X\to Y$ is a function, and if $Z\subseteq X$ is a subset, then $f\left(Z\right)=\left\{f\left(z\right)\middle| z\in Z\right\}$ is a set.

In this case, given $A,B\subseteq X$, then $f\left(A\cap B\right)$ is a set.

Answer 2

$f(A\cap B) = \{f(x) : x \in A\cap B \}$

Answer 3

It is probably like this (although the context is necessary to be absolutely sure):

• $f$ is a function that takes a set as input (for example a measure).
• $A$ and $B$ are sets
• $A \cap B$ is the intersection which is a set itself (potentially the empty set).

It can also be that $f$ is any function and it is the set of all values returned by inputting all values of $A\cap B$ into $f$, i.e. $$ f(A\cap B)=\{f(x)\mid x \in A \cap B\}.$$