For proving symmetry in E,
Given that if(a,b)∈R and (b,a)∈R ,then (a,b)∈E.
Now let (a,b) and (b,a) both belong to R then we can include (a,b) and also (b,a) in E. How?
Because for (b,a) to be included in E the condition is (b,a)∈R and (a,b)∈R. We have already assumed that both are there in R.
So conclusion is if (a,b) and (b,a) both are in R then (a,b),(b,a) are in E which maintains symmetry.
Now suppose (a,b)∈ R and (b,a) ∉ R then (a,b) ∉ E.
Similarly (b,a) can't be there in E.
Symmetric Relation E means that "if" (a,b)∈E "then" (b,a)∈E.
p->q form where p: (a,b)∈E ; q: (b,a)∈E.
=> if (a,b)∉ E (p=False ) then p->q is (False implies anything is True) True which means symmetry is there.
So if (a,b) itself is not in in E then we don't need to check for (b,a) in E.