Consider the set $A = \left \{ 1,2,3,4,..... ,n \right \}$
if $R$ is a relation defined on $A$ then $R^{-1}$ is the relation which is obtained by flipping the tuples in $R$
example: Let $R = \left \{ (1,2),(3,4),(3,1) \right \}$ then $R^{-1} = \left \{ (2,1),(4,3),(1,3) \right \}$
(i) Statement 1: If $R$ is reflexive then $R \cap R^{-1}$ is not empty
Proof : If $R$ is reflexive then $(x,x) \in R$ $\forall _{x} \in A$
$R^{-1}$ will then also contain all the reflexive pairs
Hence $R \cap R^{-1}$ is guaranteed to contain all reflexive pairs hence it will be reflexive, so this is TRUE
(ii) Statement 2: If $R$ is symmetric then $R = R^{-1}$ is not empty
Proof : If $R$ is symmetric then $(x,y),(y,x) \in R$
$R^{-1}$ will then contain $(y,x),(x,y)$ $\equiv$ $R^{-1}$ will then contain $(x,y),(y,x)$
$\therefore R = R^{-1}$, hence this statement is also TRUE
More properties of inverse relations : https://proofwiki.org/wiki/Inverse_Relation_Properties