Prove that if R is an equivalence relation on a set A, then the inverse relation R−1 is an equivalence relation on A. step by step please

Let A = {-5, -4, -3, -2, -1, 0, 1, 2, 3, 4} and define a relation R on A as follows: For all x, y EA, x Ry⇒ 3|(x - y). It is a fact that R is an equivalence relation on A. Use set-roster notation to write the equivalence classes of R. [0] [1] II [3] = [2] = = How many distinct equivalence classes does R have? classes List the distinct equivalence classes of R. (Enter your answer as a comma-separated list of sets.)

Theorem: Let R ⊆ A × A be a relation. Then R is transitive if and only if R ◦ R ⊆ R. Prove Theorem: show that R is transitive if and only if R ◦ R ⊆ R. No handwritten please

(a) Define the following terms: (i) Cartesian product of two sets; (ii) relation on a set X. (b) Write down a relation on the set {1,2,3} which is reflexive and symmetric but not transitive. (c) Let S be the relation on the set R \ {0} defined by xSy if and only if y/x € Q. Prove that S is an equivalence relation.

Prove or disprove the following statements