# Composition of equivalence relations (and union)

As part of a proof I have to show that: For $$\phi, \psi$$ equivalence relations ($$\in Eq(A)$$):

$$\phi \cup (\phi \circ \psi ) \cup (\phi \circ \psi \circ \phi )\cup (\phi \circ \psi \circ\phi \circ \psi ) \cup \ldots$$ Is also an equivalence relation.

Reflexivity is pretty simple, since for all $$a \in A: (a,a)\in\phi$$ . But I have troubles showing symmetry and transitivity...

Tanks for the help!

Let $$\theta:=\phi\cup(\phi\circ\psi)\cup(\phi\circ\psi\circ\phi)\cup\dots$$.
Assume $$x\,\theta\,y$$, then there are elements $$z_i$$ such that $$x\,\phi\,z_1\, \psi\, \dots\, z_n\, \rho\,y$$ where $$\rho$$ is either $$\phi$$ or $$\psi$$, according to the parity of $$n$$.
Even if $$n$$ is odd (whence $$\rho=\psi$$), we can make it even by adding $$z_{n+1}:=y$$ and noting that $$y\,\phi\,y$$.
But then, we get $$y\,\phi\,z_n\,\psi\,\dots\,z_1\,\phi\,x$$ showing that $$y\,\theta\, x$$.
Transitivity can be proved similarly, perhaps reducing to the case when the above $$n$$ is odd eases it up.