  A family $\mathcal{F}$ is $t$-$\it{intersecting}$ if any two members have at
least $t$ common elements. Erd\H os, Ko, and Rado proved that the maximum size
of a $t$-intersecting family of subsets of size $k$ is equal to $ {{n-t}
\choose {k-t}}$ if $n\geq n_0(k,t)$. Alon, Aydinian, and Huang considered
families generalizing intersecting families, and proved the same bound. In this
paper, we give a strengthening of their result by considering families
generalizing $t$-intersecting families for all $t \geq 1$. In 2004, Talbot
generalized Bollob\'{a}s's Two Families Theorem to $t$-intersecting families.
In this paper, we proved a slight generalization of Talbot's result by using
the probabilistic method.
