Let A, B and C be sets and let T be a set operator. Prove that $(ATB) × C = (A × C)T(B × C)$

Question

This is one of the questions on my homework.

Let A, B and C be sets and let T be a set operator. T can be Union, Intersection, Difference and Simetric Difference.

Prove that:

$(ATB) × C = (A × C) T (B × C)$

I don't really know how to handle ×.

If it was to prove, say:

$ C∪(A∩B) = (C∪A)∩(C∪B)$

I would know how to handle it. However, the × was not really defined on the question.

I tried searching for × math symbol. But it was described as ×, a variable, or the symbol for multiplication.

I also tried to find a question similar to this one on Stack Exchange. However, I couldn't find it.

How can I prove this? What is the first step?

Answer 1

$\def\inv{^{-1}}$

Hint: In general, let $f: X \to Y$ be any map and for $A \subseteq Y$, define $f\inv(A) = \{ x \in X : f(x) \in A\}$. Show that $f\inv$ respects the set operations of union, intersection and complement, and of course other operations derived from these. For example $$ f\inv(A \cup B) = f\inv(A) \cup f\inv(B). $$

Can you derive your desired statement(s) from this principle?