Let Z be the set of all integers andA = {(x, y); x4 – y4 = 175, x, y ∈ Z}B = {(x, y); x > y, x, y ∈ Z}Then, the number of elements in A ∩ B is

To find the number of elements in A ∩ B, we need to find the pairs (x, y) that satisfy both conditions A and B.

Condition A: x^4 - y^4 = 175 Condition B: x > y

First, let's solve the equation x^4 - y^4 = 175. This can be factored into (x^2 - y^2)(x^2 + y^2) = 175.

Since x and y are integers, (x^2 - y^2) and (x^2 + y^2) must also be integers. Therefore, we need to find the pairs of integers that multiply to 175. These pairs are (1, 175), (-1, -175), (5, 35), and (-5, -35).

For (1, 175), we have x^2 - y^2 = 1 and x^2 + y^2 = 175. Solving these equations simultaneously, we get two pairs of solutions: (x, y) = (12, 11) and (-12, -11).

For (-1, -175), we have x^2 - y^2 = -1 and x^2 + y^2 = -175. However, x^2 + y^2 cannot be negative, so there are no solutions in this case.

For (5, 35), we have x^2 - y^2 = 5 and x^2 + y^2 = 35. Solving these equations simultaneously, we get two pairs of solutions: (x, y) = (6, 1) and (-6, -1).

For (-5, -35), we have x^2 - y^2 = -5 and x^2 + y^2 = -35. Again, x^2 + y^2 cannot be negative, so there are no solutions in this case.

So, the pairs (x, y) that satisfy condition A are (12, 11), (-12, -11), (6, 1), and (-6, -1).

Now, let's consider condition B: x > y. The pairs that satisfy this condition are (12, 11) and (6, 1).

Therefore, the number of elements in A ∩ B is 2.