A relation R is defined on R by xRy if x-y e Z. Prove that R is an equivalence relation. Determine the equivalence class of 0. Determine the equivalence class of 3. Describe all distinct equivalence classes for R.

Let A={1,2,3,4,5,6} and consider the following 3 subsets of A. A, = {1,3,5}; A2 = %3D {2,4, 6}; A3 = {3, 6}. Define the relation R on set A as follows: x R y if and only if { x, y} C A or { x, y}C A, or { x,y}C A. That is, x R y if and only if x and y are both in A, or x and y are both in A2 or x and y are both in A3.

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.

3. Let R be a relation defined on the set A = N × N where (a, b) R (c, d) if að = cd. Show R is an equivalence relation. Then, list the elements of the equivalence class [(9, 2)].