Function "applied on" a set of sets.

Question

Let $f\colon X\to Y$ and $A\subset X$. Halmos writes, in his book "Naive Set Theory", that $f(A)$ denotes the image of $A$ under $f$; $f(A):=\{y\in Y: \exists x \in A, f(x)=y\}$. If we have $B\subset X$, is it then typical to use $f(\{A,B\}):=\{f(A),f(B)\}$?

Answer 1

More generally if $\mathcal A\subseteq\wp(X)$ then we could state that $f(\mathcal A)=\{f(A)\mid A\in\mathcal A\}$.

And we can go steps further in this.

However IMHV it is wise not to introduce such notation without making clear to the reader what you mean.

Answer 2

Whenever you have a function $f\colon X\to Y$, it induces function $\bar f\colon \mathcal P(X)\to \mathcal P(Y)$ by $\bar f(A) := \{ f(x)\mid x\in A \}$.

Usually, we just write $f(A)$ for $\bar f(A)$ when there is no possible confusion, but what you suggest would be very confusing, and frankly I don't see a point. Note that some authors use $f[A]$ instead of $f(A)$.

But, your idea is right in the sense that the above $\bar f$ induces a new function $\ \bar{\!\!\bar f}\colon \mathcal P(\mathcal P(X))\to \mathcal P(\mathcal P(Y))$. If you wanted to iterate the process in this way, I would suggest using this or any other notation to distinguish between these induced functions and the original one, for the sake of clarity of what is input and what is output.

Writing $f(A)$ for $\bar f(A)$ would be an example of what we call abuse of notation. There is a reason for the word abuse, it is to give a warning that you shouldn't overdo it if you want readers to understand what you write.