2011 AMC 10A Problem 6

The union $A \cup B$ is everything that is in $A$, in $B$, or in both. We want it as small as possible, so we want $A$ and $B$ to share as much as they can.

1. It can't be smaller than 20. The union contains every element of $A$, and $A$ has $20$ elements. So the union has at least $20$.
2. 20 can happen. Make every element of $B$ also an element of $A$. For example, let $A$ be the numbers $1$ to $20$ and $B$ be the numbers $1$ to $15$. Then the union is just $1$ to $20$, which has $20$ elements.

The smallest possible union has $20$ elements.

The answer is $\boxed{\textbf{(C)}\ 20}$.

The number of elements in the union is $20 + 15$ minus the number of elements the two sets share (otherwise the shared ones are counted twice). The two sets can share at most $15$ elements, since $B$ only has $15$. Sharing all $15$ gives the smallest union, $20$. Sharing none gives the largest, $35$. Questions about the smallest or largest union usually come down to these two extreme pictures: one set inside the other, or no overlap at all.

The union contains all of $A$, so it has at least $20$ elements.

Exactly $20$ is achievable: let $B$ consist of $15$ of the elements of $A$. Then $A \cup B = A$, which has $20$ elements.

The answer is $\boxed{\textbf{(C)}\ 20}$.