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(1) R is reflexive, because if x∈ℤ, x−x=0 that is even, and (x,x)∈R.
R is symmetric, because if x, y∈ℤ such that (x,y)∈R, x−y is even, say 2n, and y−x=−2n=2(−n) that is even too: (y,x)∈R.
R is transitive, because if x, y, z∈ℤ such that (x,y)∈R and (y,z)∈R, x−y=2n and y−z=2m, x−z=x−y+y−z=2n+2m=2(n+m), that is even: (x,z)∈R.
(2) If x is even, x=2n. (x,y)∈R iff 2n−y is even iff y is even. So, [x] is the subset of even numbers, and the other class is the subset of odd numbers.
(3) 0 is a representative of the subset of even numbers, 1 is a representative of the subset of odd numbers.