What is beta equivalence? In the script I am currently reading on the lambda calculus, beta equivalence is defined as this:


  The $\beta$-equivalence $\equiv_\beta$ is the smallest equivalence that contains $\rightarrow_\beta$.


I have no idea what that means. Can someone explain it in simpler terms? Maybe with an example?

I need it for a lemma following from the Church-Russer theorem, saying


  If M $\equiv_\beta$ N then there is a L with M $\twoheadrightarrow_\beta$ L and N $\twoheadrightarrow_\beta$ L.
$\to_\beta$ 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.