If $P \leq G$, $Q\leq G$, are $P\cap Q$ and $P\cup Q$ subgroups of $G$? 
$P$ and $Q$ are subgroups of a group $G$. How can we prove that $P\cap Q$ is a subgroup of $G$? Is $P \cup Q$ a subgroup of $G$?

Reference: Fraleigh p. 59 Question 5.54 in A First Course in Abstract Algebra.

Counter: Letting $x = \left( \textrm{1 2 3}  \right), y = \left( \textrm{1 2}  \right)$. Then $\langle x \rangle , \langle y \rangle$ are subgroups of $S_3$, but their union is not (it is not closed).  
Intersection: if $x, y \in H \cap K$, then $x, y \in H, K$. From this, closedness, existence of inverses and identity in $H, K$ follow easily, and so in $H \cap K$.
