A relation on a set A is circular iff , y, z x~y and y~z implies z~x. Prove that a relation is an equivalence relation iff it is circular and reflexive.

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

Let S = {a, b, c}. Then R = {(a, b)} is a relation on S. Which of the properties reflexive, symmetric andtransitive does the relation R possess? I know that its not reflexive or symmetric but its vacuously transitive, but I dont get why, so please explain step by step

Let R be the relation on the set of ordered pairs of positive integers such that ((a, b), (c,d)) = R if and only if ad = bc. Show R is an equivalence relation.