Question

Let R be the relation on the integers defined by arb iff 3 divides (a-b).

(a) Is R reflexive?
(b) Is R symmetric?
(c) Is R transitive?
(d) Is R an equivalence relation?
a) True
b) False

Answer 1

The relation R is reflexive, symmetric, transitive, and an equivalence relation.

Step-by-step explanation:

a) The relation R is reflexive if every integer a is related to itself. In this case, arb if 3 divides (a-b). So if we let a = b, we have to check if 3 divides (a-a) which is equivalent to 3 dividing 0. Since 3 divides 0, R is reflexive.

b) The relation R is symmetric if whenever arb, then bra. In this case, arb if 3 divides (a-b). So if we have arb, we need to check if bra. Since 3 divides (a-b), then -3 divides (b-a), which means bra. Hence, R is symmetric.

c) The relation R is transitive if whenever arb and brc, then arc. In this case, arb if 3 divides (a-b). So if we have arb and brc, we need to check if arc. Since 3 divides (a-b) and 3 divides (b-c), it follows that 3 divides (a-c). Hence, R is transitive.

d) A relation is an equivalence relation if it is reflexive, symmetric, and transitive. From the explanations above, we can see that R is reflexive, symmetric, and transitive. Therefore, R is an equivalence relation.