How many equivalence relations on A=|42| are there that have distinct equivalence classes of size 4, 7,7, 8,8, and 8?

Answer the questions a) How many binary relations exist on an empty set? b)

Define the following relation R in the set X = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9}. For integers x, y ∈ X, let xRy if and only if x^2 - y^2 or x^2 + y^2 is divisible by 10. Prove that R is an equivalence relation and determine the equivalence classes of R.

Let R = {(a, b) | a = b (mod 3)} be an equivalence relation on the set of integers Z. What are the three equivalence classes of R? Provide five integers that are members of each equivalence class.