Closure of equivalence relations Show that the transitive closure of the symmetric closure
of the reflexive closure of a relation R is the smallest
equivalence relation that contains R.
I can understand the statement intuitively but can't come up with a mathematical proof

HINT: Let $S$ be the transitive closure of the symmetric closure of the reflexive closure of $R$. You have to show three things:


*

*$R\subseteq S$.  

*$S$ is an equivalence relation.  

*If $E$ is an equivalence relation containing $R$, then $E\supseteq S$.


The first of these is pretty trivial, and the second isn’t very hard: just show that the symmetric closure of a reflexive relation is still reflexive, and that the transitive closure of a symmetric, reflexive relation is still symmetric and reflexive. For (3), show that every ordered pair in $S$ must necessarily belong to any equivalence relation containing $R$.
