Solution 1
Given an instance $\Pi$ of set cover with non-empty subsets of size at most 2, consider two instances of set cover with subsets of size exactly 2:
- $\Pi'$ contains all subsets in $\Pi$ of size exactly 2.
- $\Pi''$ contains all subsets in $\Pi$ of size exactly 2, and in addition, for each singleton $\{x\} \in \Pi$, the subset $\{x,z^*\}$, where $z^*$ is a new element (the same element is used for all singletons).
If an optimal solution for $\Pi$ doesn't use any singletons, then it will also be an optimal solution for $\Pi'$. Otherwise, it will be an optimal solution for $\Pi''$.
Solution 2
Let $\Pi$ be an instance of set cover with non-empty subsets of size at most 2. Say that a singleton $\{x\} \in \Pi$ is covered if $\Pi$ contains some set $\{x,y\}$. Any solution for $\Pi$ which uses $\{x\}$ could instead use $\{x,y\}$, so we can remove all covered singletons from $\Pi$. All singletons that remain in $\Pi$ must participate in any set cover, so we can set them aside, obtaining an instance $\Pi'$ in which all subsets have size exactly 2. Given an optimal solution for $\Pi'$, we obtain an optimal solution for $\Pi$ by adding all uncovered singletons that we had set aside.
