What is beta equivalence?
$\to_\beta$ is the one-step relation between terms in the $\lambda$-calculus. This relation is neither reflexive, symmetric, or transitive. The equivalence relation $\equiv_\beta$ is the reflexive, symmetric, transitive closure of $\to_\beta$. This means 


If $M\to_\beta M&#39;$ then $M\equiv_\beta M&#39;$. 
For all terms $M$, $M\equiv_\beta M$ holds.
If $M\equiv_\beta M&#39;$, then $M&#39;\equiv_\beta M$.
If $M\equiv_\beta M&#39;$ and $M&#39;\equiv_\beta M&#39;&#39;$, then $M\equiv_\beta M&#39;&#39;$.
$\equiv_\beta$ is the smallest relation satisfying conditions 1-4.


More constructively, first apply rules 1 and 2, then
repeat rules $3$ and $4$ over and over until they add no new elements to the relation.