Let R be a reflexive and transitive relation on a set A. Define a new relation E on A as

 E = {(a,b) ∣ (a,b)∈R and (b,a)∈R} 

(a) Prove that E is an equivalence relation on A.
(b) Define a reason ≤ on the equivalence classes of E as E₁ ≤ E₂ if ∃a,b such that a∈E₁, b∈E₂ and (a,b)∈R. Prove that ≤ is a partial order.