Let $R$ and $S$ be two relations on a set $A.$ Consider the following statements: 

$S_1:R$  and  $S$ are transitive, then  $R\cup S$ is also transitive 
$S_2:R$  and  $S$ are transitive, then $R\cap S$  is also transitive 
$S_3:R$  and $S$ are reflexive, then  $R\cap S$ is also reflexive 
$S_4:R$  and  $S$ are symmetric then $R\cup S$  is also symmetric 
Then correct statements are:

Transitivity property usually not hold in unions. For example, consider the set $A=\{1,2,3\}$  and $R=\left\{(1,2)\right\},S=\left\{(2,3)\right\}$.
Clearly  R and S  are transitive but  $R\cup S=\{(1,2),(2,3)\}$ is not transitive, 
So $\left(S_1\right)$ is not true. 
For $\left(S_2\right)$ . Let  $(a,b),(b,c)\in R\cap S$
$\Rightarrow (a,b),(b,c)\in R$ and   $(a,b),(b,c)\in S$
 $\Rightarrow (a,c)\in R,(a,c)\in S\Rightarrow (a,c)\in R\cap S$
$\therefore$  If  R,S are trans., then $R\cap S$ is also trans. 
So,  $\left(S_2\right)$ is true. 
For  $\left(S_3\right)R,S$ are reflexive, so 
 $(a,a)\in R,(a,a)\in S\forall a\in A$
$\therefore \forall a\in A,(a,a)\in R\cap S$  and  $R\cup S$.
And so  $R\cup S$ and $R\cap S$  both reflexive. 
Also for $S_4,R$ and  S are symmetric. 
$\therefore (a,b)\in R\cup S\Rightarrow (a,b)\in Ror(a,b)\in S$ 
$\Rightarrow (b,a)\in R$  or  $(b,a)\in S\Rightarrow (b,a)\in R\cup S$
$\Rightarrow R\cup S$  is symmetric.