isomorphism of two semigroups

Question

Let $X$ be an arbitrary set and denote $Y = \mathcal{P}(X)$. We consider two semigroups $G_1 = (Y,\ \cup)$ and $G_2 = (Y,\ \cap)$.

I must prove that $G_1$ and $G_2$ are isomorphic.

Answer 1

Take the map $\varphi: G_1\to G_2$ mapping $S\mapsto X\setminus S$. Then, $$\varphi(S\cup T)= X\setminus(S\cup T) = (X\setminus S)\cap(X\setminus T) = \varphi(S) \cap \varphi(T)$$ so this is a morphism of semigroups. It is an isomorphism because its inverse is itself, set-theoretically.