Final answer:
The relation r is reflexive because for every integer a, 3 divides (a - a), which is 0, so the ordered pair (a, a) is in r for all integers.
Step-by-step explanation:
The question asks whether the relation r, defined on the set of integers such that arb if and only if 3 divides (a-b), is reflexive. A relation R on a set S is said to be reflexive if every element is related to itself; that is, for all a in S, the ordered pair (a, a) is in R. To check if relation r is reflexive, we must determine if for every integer a, the pair (a, a) is in r.
By the definition of relation r, arb is true if 3 divides (a - b). For reflexive property, we consider the case where a = b, which gives us (a - a) or 0. Since 3 divides 0 (any number divides 0), the relation r is reflexive because for every integer a, (a, a) satisfies the relation, meaning 3 divides (a - a) which equals to 0. To further illustrate the concept, suppose we have an integer 5. In this relation, (5, 5) would be considered part of r because 3 divides (5 - 5) = 0 without any remainder. This example confirms that the relation r is reflexive as it meets the necessary conditions for all integers.