Prove or disprove: If $R$ and $S$ are two equivalence relations on a set $A$, the $R \cup S$ is also an equivalence relation on $A$.
This statement has been disproven here. In addition, we know that $R \cap S$ is indeed an equivalence relation through simply mathematical reasoning. My question is: Is there any intuition for why $R\cup S$ would not itself be an equivalence relation as well? If there is, then what is the intuition?
This question came to mind after seeing that $R\cap S$ is and equivalence relation on $A$ but not $R \cup S$.
Any help in the intuition part is great! Thanks!