Final answer:
The relation r, defined by x - y < 1 for integers, is reflexive but neither symmetric nor transitive. Therefore, it is not an equivalence relation. The correct characterization is that r is not an equivalence relation.
Step-by-step explanation:
The question pertains to the properties of a mathematical relation defined on the set of integers where x is related to y if x - y is less than 1. To characterize this relation, we must examine if it meets the criteria for being reflexive, symmetric, and transitive, which are necessary conditions for a relation to be an equivalence relation.
Firstly, a relation is reflexive if every element is related to itself. In this case, for any integer x, x - x = 0, which is indeed less than 1, hence the relation r is reflexive.
Secondly, to be symmetric, if x is related to y, then y must also be related to x. Here, if x - y < 1, it does not necessarily follow that y - x < 1 (consider x = 2 and y = 0 as a counterexample, where 2 - 0 < 1 but 0 - 2 is not less than 1). Therefore, the relation r is not symmetric.
Lastly, a transitive relation means if x is related to y and y is related to z, then x must be related to z. However, if x - y < 1 and y - z < 1, it cannot be assured that x - z < 1 (consider x = 2, y = 1, and z = 0). Therefore, the relation r is not transitive.
Since r is not symmetric and not transitive, it cannot be an equivalence relation. Therefore, statement 4 correctly characterizes the relation r—that it is not an equivalence relation, although it is reflexive.