Problem with equivalence relations.

Question

Proof that if $\mathrm{R}_1$ and $\mathrm{R}_2$ are equivalence relations then new relation $\mathrm{R}$ which is determined as $(a\,\mathrm{R}\,b)\ (a\,\mathrm{R}_1\,b\wedge a\,\mathrm{R}_2\,b)$ is also equivalence relation.

Answer 1

1) Refexive. Since $(x\,\mathrm{R}_1\,x)$ and $(x\,\mathrm{R}_2\,x)$, $\dots$

2) Symmetric. If $(x\,\mathrm{R}\,y)$, then $(x\,\mathrm{R}_1\,y)$ and $(x\,\mathrm{R}_2\,y)$. Thus $\dots$

3) Transitive. If $(x\,\mathrm{R}\,y)$ and $(y\,\mathrm{R}\,z)$, then $\dots$