Symmetric closure

Question

Let be a relation on a non-empty set . Define, explicitly in terms of , the relation which is the symmetric closure of .

i just started on relations and is confused by how do i go about proving this question in terms of R.

Answer 1

If $R \subset A \times A$ is a relation on $A$ then we have that the symmetric closure of $R$ is $$\bar{R} = R \cup \{(y, x): (x, y) \in R\}.$$ So you just add the mirror images of all the relators (if they are not already elements of the relation). To prove that this is indeed the symmetric closure, first make the easy observation that the relation is symmetric now. Moreoever, any other symmetric relation containing $R$ also contains the mirror images of $R$ (by symmetry). Since we have added nothing else, we get the other inclusion.