7. This question involves an if and only if proof, so you will need to prove the statement in both directions.
Firstly, assume that the relation R is symmetric and prove that R = R-1.
Proof: Since R is symmetric, for all x and y in the set A, if (x, y) is in R, then (y, x) is also in R. We can consider the ordered pairs in R as occuring as singles (a,a) or as couples (a,b) and (b,a). If (a,a) occurs in R, then (a,a) will also occur in R-1. If (a,b) and (b,a) occur in R, then (b,a) and (a,b) occur in R-1. Thus any ordered pair belonging to the relation R will also belong to the relation R-1 (and vice versa). Hence R = R-1.
Secondly, assume that R = R-1 and prove that R is symmetric.
Proof: Since R = R-1, we know, for any elements a and b in the set A, if the ordered pair (a, b) occurs in R then the ordered pair (b, a) must also occur in R, since it must occur in R-1 and R = R-1. Thus, by definition, R is symmetric.