A relation R is defined on Z by a R b if a³ ≡ b³( mod 5). Prove that R is an equivalence relation. Determine the distinct equivalence classes.

Let S be the set of all students at FIU. Determine whether the relation R on S (defined below) is reflexive, irreflexive, symmetric, antisymmetric and transitive? Support your answer with a proof or a counterexample per each property. R = {(x, y) = SxSx has a better GPA than y} alternatively, we can write Ry if student x has a better GPA than student y

#1. Let R be the relation defined on the set of all integers Z as follows: for all integers m and n, m R n e= m - n is divisible by 5. Prove that R is Equivalence Relation.

18. Let R be the relation on R defined by xRy → x – y is an integer. Prove that R is an equivalence relation.

8. Let A be a set of nonzero integers and let R be a relation on A × A defined by (a, b)R(c, d) whenever ad = bc. Show that R is an equivalence relation. That is, R is reflexive, symmetric, and transitive.