I am trying to figure out how to show $E \circ E=E$ (the composition of two equivalence relations is an equivalence relation). I have already shown that $a(E \circ E)b \implies aEb$ and need to figure out how to show that $aEb \implies a(E \circ E)b$. I know that I need to show what allows me to go from $aEb$ to $\exists z\ (aEz \land zEb)$ but I'm not sure how to get there. Thanks in advance!
2026-04-08 14:32:41.1775658761
Prove that $E \circ E=E$ (the composition of two equivalence relations is an equivalence relation)
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Assume $aEb$. Thus $aEa$ anb $aEb$, whence $a E\circ E b$.