The first part is asking you to show that P is a partition of of set A- that is, that every element of A is in one and only one set in P. Let x be an element of A. Since P1 is a partition of A, there is a unique set in P1 that contains x. Since P2 is a partition of A, there is a unique set in P2 that contains x. x is in and only that set in P.

The "equivalences" defined by P1 and P2 are: x≡1 y if and only if x and y are in the set in P1 and x≡2 y if and only if x and y are in the same set in P2. "x ≡ y" if and only if x and y are in the same set in P. Since P is made of intersections of sets in P1 and P2, in order to be in an intersection of two sets, x and y would have to both be in the same set in P1 and P2: x ≡ y if and only if x≡1 yandx≡2 y.