Bipartite graph minimal amount of vertices required
Let's define a graph $G=(V,E)$, for which:


Set of vertices $V$ is identical to your SET1.
Two vertices $v_1,v_2 \in V$ are connected by an edge, if and only if there exists a vertex from the SET2, incident to both $v_1$ and $v_2$ in your bipartite graph.


For instance, the graph below corresponds to the bipartite graph from the example in your question:



Now, your problem is reduced to the Minimum Vertex Cover problem on the graph $G$, defined above, so you'll be able to use known algorithms to find minimum vertex cover in your SET1 only.

This graph $G$ doesn't have any special structure, which could be used to find its minimum vertex cover more effectively than in general case. Also, any graph can be transformed to your bipartite graph form by splitting each edge by a new vertex. So, it means that your problem is equivalent to the minimum vertex cover problem, which is NP-complete - thanks to @narekBojikian for mentioning that in his answer.