Prove that the intersection of two subgroups is a subgroup. In more detail, if  $G$  is a group and $H_1$, $H_2$   are subgroups of G   then $H_1 \cap H_2$   is a subgroup of  G.
Next, give an example of a particular group $G$  (any one you like), and two different subgroups  $H_1$, $H_2$ of $G$  , compute the intersection $H_1 \cap H_2$  , and verify it is indeed a subgroup. 
Finally, give three examples showing that $H_1 \cup H_2$   need not be a subgroup of $G$ .

Let $G=S_3=\{(1),(1,2),(1,3),(2,3),(1,2,3),(1,3,2) \}$ and 
$$H_1=\{(1),(1,3) \}$$
$$H_2=\{(1),(1,2) \}$$
$$H_1\cap H_2=\{(1) \} $$
$$H_1\cup H_2=\{(1),(1,2),(1,3)\} $$
The union is not group since it is not closed as $(1,2)(1,3)=(1,3,2)\notin H_1\cup H_2$
for proof of inersection is a subgroup,just write; if you can take an element from a set you can do it,and follow the the other answers.
