With relations $R, E$ on a set $Y$ (so that: $R, E \subseteq Y \times Y$) prove or disprove: If $R$ and $E$ are transitive, then so is $R \cap E.$

Question

How would I go about solving this question?

I can give examples of $R$ and $E$ being transitive, but if there is an intersection between the two sets, then some information is missing to make it transitive?

Answer 1

Well, if $R$ and $E$ are transitive, then $R\cap E$ is transitive.

Indeed, let $(a,b), (b,c)\in R\cap E$. Then $(a,b),(b,c)\in R$ and $(a,b),(b,c)\in E$. Since $R$ and $E$ are transitive, $(a,c)\in R$ and $(a,c)\in E$. Hence, $(a,c)\in R\cap E$.