Union is Distributive over Intersection with Indexed families of sets Proof Explanation using finite trivial example

Question

What does the notation $F_{a,b}$ mean? Is $F_{a,b}$ a set?

What does F$_{a,f(a)}$ mean? Is this F$_{a,f(a)}$ a set?

I have no idea how the lh and rh equality holds. Can I get a finite trivial example of when this equality shown above holds where we define or instantiate some example the set A, B, define what $ B^{A}$ is and define what F$_{a,f(a)}$ is?

Answer 1

The notation $f \in B^A$ is just that f is an element in the set of all possible functions from $A \to B$.

Then I assume that the set $F_{a,b}$ is the set of all functions such that $f(a) = b$

All the equality follows from the definition of a function, if you see the condition given is equivalent to the definition of a function in terms of binary relations.

Then use as an example the set $A = \{0,1\}, B = \{2,3\}$ and construct all the possible functions from this set (we will have $|B|^{|A|} = 2^2$ functions) and try to prove this equalities.