Determine the following relation is an equivalence relation or not. If the relation is an equivalence relation, describe the partition given by it. x~y in R if |x-y|<4

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The relation must be reflexive, symmetric, and transitive to be an equivalence relation.

Let's consider our relation: x~y in R if |x-y|<4.

(i) It's reflexive, because |a-a|=0<4 hence a~a in R.

(ii) It's symmetric, because |x-y|=|y-x|. Hence if |x-y|<4 then |y-x|<4, if x~y then y~x.

(iii). It's not transitive, because 1~4 (|1-4|=3<4), 4~7 (|4-7|=3<4), but |1-7|=6>4, 1 is not related to 7.

So the relation x~y in R if |x-y|<4 is not equivalence relation.