It is proved that the elements in the left set belong to the right set: if y belongs to f(AUB), then x belongs to AUB, so that f(x)=y, then x belongs to a or b, that is, y belongs to f(A) or f(B), that is, it is proved that f(AUB) is contained in f(A)Uf(B).
On the other hand, if Y belongs to f(A)Uf(B), then Y belongs to f(A) or F (b); If y belongs to f(A), then x belongs to a, so f(x)=y, and further x belongs to AUB, that is, y belongs to f(AUB).
It is proved that the elements in the left set belong to the right set: if Y belongs to f(A∩B), then X belongs to A∩B, so that f(x)=y, then X belongs to A and B, that is, Y belongs to f(A) and f(B), which proves that f(A∩B) is included in F.
On the contrary, as a counterexample, if F is constant after acting on any X, such as f(x)= 1, the intersection of A and B is an empty set, but f(A)∩f(B)= 1.