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Let p subset N x N such that p(x, y) <=> x+y is even. Then we have
[0]p = {0, 2, 4, 6, 8, 10, ...}, [1]p = {1, 3, 5, 7, 9, 11, ...}, [2]p = {0, 2, 4, 6, 8, 10, ...}, [3]p = {1, 3, 5, 7, 9, 11, ...}, [4]p = {0, 2, 4, 6, 8, 10, ...}, ...
We see that [0]p union [1]p = N and [0]p intersection [1]p = 0.