True or False:

If H and K are subgroups of G, then $\displaystyle H \cap K$ is a subgroup of G.

If $\displaystyle H_{t}$ is a subgroup of G for all $\displaystyle t \in I$, then $\displaystyle \cap_{t \in I} H_{t}$ is a subgroup of G.

Explanations would be greatly appreciated.