Let R and S be binary relations on a set A. Solve for each case: If R and S are i) reflexive, ii) symmetric, iii) transitive, what would R∪S be?

Question

I had a similar question just now and it peaked my interest on binary relations. Solving i)

"Assuming R and S are reflexive so: Let x ∈ A. Then (x, x) ∈ R and (x, x) ∈ S (reflexive property of R and S). R ∪ S would then be reflexive as well, since (x,x) ∈ R ∪ S."

Does that make any sense? As in, is this correct to assume? How would one go about proving it for R ∩ S? If R ∩ S = (x,x) it'd also be reflexive, wouldn't it? What about the symmetry and transitivity of R ∩ S?