Proof that isomorphic graphs must have the same number of vertices

Question

An isomorphism of graphs G and H is a bijection between the vertex sets of G and H. So I know that from definition of a bijection number of vertices must be the same, but how to describe it offically on a test ?

Answer 1

You just proved it.

Anyway, this is a special case of a more general phenomenon, namely that:

• functors preserve isomorphisms
• hence functors preserve isomorphicness.

In this case, we're talking about the underlying set functor $$U : \mathbf{Grph} \rightarrow \mathbf{Set}.$$