Question: Let A = {1, 2, 3, 4}. For any function f : A A and any relation R on A, we define the relation S

Let A = {1, 2, 3, 4}. For any function f : A A and any relation R on A, we define the relation S on A by: for any a, b A, aSb if and only if (f (a), f (b)) R.

For each of the following statements, prove or disprove the statement.

(a) For all functions f : A A and all relations R on A, if S is reflexive then R is reflexive.

(b) For all functions f : A A and all relations R on A, if R is reflexive then S is reflexive.

(d) For all functions f : A A and all relations R on A, if R is symmetric then S is symmetric.

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