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R := {<x, y>: x in N /\ y in N /\ x <= y}
R is a relation on N (R subset N x N); it is also a relation on Z (R subset Z x Z).
S := {<x, x/2>: x in N}
S is a relation between N and Q (R subset N x Q) and it is also a relation on Q (R subset Q x Q); it is not a relation on N (R not subset N x N).