Question

{F} Let R1 and R2 be symmetric relations. Is R1 ∩ R2 also symmetric? Is R1 ∪ R2 also symmetric?

Answer 1

Let $R_1$ and $R_2$ be symmetric relations. Let us prove that $R_1 ∩ R_2$ is also symmetric. Let $(a,b)\in R_1\cap R_2.$ Then $(a,b)\in R_1$ and $(a,b)\in R_2.$ Since $R_1$ and $R_2$ are symmetric relations, $(b,a)\in R_1$ and $(b,a)\in R_2.$ It follows that $(b,a)\in R_1\cap R_2,$ and hence $R_1 ∩ R_2$ is also symmetric relation.

Let $R_1$ and $R_2$ be symmetric relations. Let us prove that $R_1 \cup R_2$ is also symmetric. Let $(a,b)\in R_1\cup R_2.$ Then $(a,b)\in R_1$ or $(a,b)\in R_2.$ Since $R_1$ and $R_2$ are symmetric relations, $(b,a)\in R_1$ or $(b,a)\in R_2.$ It follows that $(b,a)\in R_1\cup R_2,$ and hence $R_1 \cup R_2$ is also symmetric relation.