Let A={1,2,3}
R is a relation on set A which is reflexive and transitive.
R:{(1,1),(2,2),(3,3),(1,2)} → this relation is reflexive and transitive but not symmetric.
Now a new relation E has been defined on set A.
E = {(a, b) | (a, b) ∈ R and (b, a) ∈ R}
E:{(1,1),(2,2),(3,3)} → E will always be a identify relation on set A.
Identify relation is always reflexive, symmetric and transitive.
→ So, E is an equivalence relation on set A.
For second part:
There will be 3 equivalence classes E1, E2, E3.
H={E1,E2,E3}
a relation ≤ on the equivalence classes of E as E1≤E2 if ∃a,b such that a∈E1,b∈E2 and (a,b)∈R.
E1={1} , E2={2}, E3={3}
∃1,2 such that 1∈E1, 2∈E2, and (1,2)∈R. So, E1≤E2
New relation will be reflexive :
→ ∃1,1 such that 1∈E1, 1∈E1, and (1,1)∈R. So, E1≤E1
→ ∃2,2 such that 2∈E2, 2∈E2, and (2,2)∈R. So, E2≤E2
→ ∃3,3 such that 3∈E3, 3∈E3, and (3,3)∈R. So, E3≤E3
New relation will be antisymmetric and transitive.
So, we can say new relation is a partial order.