Let A = {1,2,3,4}. For any function f from A to A and any relation R on A, we define a relation S on A by: For all a, b A, aSb f(a)Rf(b). Prove or disprove each of the following statements. (a) For every function f from A to A and every relation R on A, if R is reflexive, then S is reflexive. (b) For every function f from A to A and every relation R on A, if S is reflexive, then R is reflexive. (c) For every function f from A to A and every relation R on A, if R is symmetric, then S is symmetric. (d) For every function f from A to A and every relation R on A, if S is symmetric, then R is symmetric.
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Let A = {1, 2, 3, 4}. For any function f from A to A and any relation R on A, we
define a relation S on A by:
For all a, b ∈ A, aSb ⇐⇒ f(a)Rf(b).
Prove or disprove each of the following statements.
(a) For every function f from A to A and every relation R on A, if R is reflexive,
then S is reflexive.
(b) For every function f from A to A and every relation R on A, if S is reflexive, then
R is reflexive.
(c) For every function f from A to A and every relation R on A, if R is symmetric,
then S is symmetric.
(d) For every function f from A to A and every relation R on A, if S is symmetric,
then R is symmetric.
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