a)Let R be a relation on the set Z × Z defined by the following rule: For all (a, b),(c, d) ∈ Z × Z, (a, b) R (c, d) if and only if 3a + 2c = 3b + 2d. Is R reflexive ? Answer with ‘YES’ or ‘NO’. If your answer is ‘YES’, give a detailed proof. If your answer is ‘NO’, provide an appropriate counterexample and explain. (b) Is R symmetric ? Answer with ‘YES’ or ‘NO’. If your answer is ‘YES’, give a detailed proof. If your answer is ‘NO’, provide an appropriate counterexample and explain. (c) Let S be a relation on the set Z, defined by the following rule: For all a, b, ∈ Z, a S b if and only if 3a + 5b is even. Is S transitive? If so, give a detailed proof. If not, provide an appropriate counterexample and explain.
a)Let R be a relation on the set Z × Z defined by the following rule:
For all (a, b),(c, d) ∈ Z × Z, (a, b) R (c, d) if and only if 3a + 2c = 3b + 2d.
Is R reflexive ? Answer with ‘YES’ or ‘NO’.
If your answer is ‘YES’, give a detailed proof. If your answer is ‘NO’, provide an
appropriate counterexample and explain.
(b) Is R symmetric ? Answer with ‘YES’ or ‘NO’.
If your answer is ‘YES’, give a detailed proof. If your answer is ‘NO’, provide an
appropriate counterexample and explain.
(c) Let S be a relation on the set Z, defined by the following rule:
For all a, b, ∈ Z, a S b if and only if 3a + 5b is even.
Is S transitive? If so, give a detailed proof. If not, provide an appropriate counterexample and explain.
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