Question

If R and S are relations on a set A, then prove that
(i) R and S are symmetric ⇒ R ∩ S and R ∪ S are symmetric
(ii) R is reflexive and S is any relation ⇒ R ∪ S is reflexive.

Answer 1

(i) R and S are symmetric relations on the set A.

$$\begin{gathered} \Rightarrow R\subset A\times A\mathrm{and}S\subset A\times A \\ \Rightarrow R\cap S\subset A\times A \\ \mathrm{Thus},R\cap S\mathrm{is}\mathrm{a}\mathrm{relation}\mathrm{on}A. \\ \mathrm{Let}a,b\in A\mathrm{such}\mathrm{that}\left(a,b\right)\in R\cap S.\mathrm{Then}, \\ \left(a,b\right)\in R\cap S \\ \Rightarrow \left(a,b\right)\in R\mathrm{and}\left(a,b\right)\in S \\ \Rightarrow \left(b,a\right)\in R\mathrm{and}\left(b,a\right)\in S\left[\mathrm{Since}R\mathrm{and}S\mathrm{are}\mathrm{symmetric}\right] \\ \Rightarrow \left(b,a\right)\in R\cap S \\ \mathrm{Thus}, \\ \left(a,b\right)\in R\cap S \\ \Rightarrow \left(b,a\right)\in R\cap S\mathrm{for}\mathrm{all}a,b\in A \\ \mathrm{So},R\cap S\mathrm{is}\mathrm{symmetric}\mathrm{on}\mathrm{A}. \end{gathered}$$

Also,
$$\begin{gathered} \mathrm{Let}a,b\in A\mathrm{such}\mathrm{that}\left(a,b\right)\in R\cup S \\ \Rightarrow \left(a,b\right)\in R\mathrm{or}\left(a,b\right)\in S \\ \Rightarrow \left(b,a\right)\in R\mathrm{or}\left(b,a\right)\in S\left[\mathrm{Since}R\mathit{}\mathrm{and}S\mathrm{are}\mathrm{symmetric}\right] \\ \Rightarrow \left(b,a\right)\in R\cup S \\ \mathrm{So},R\cup S\mathrm{is}\mathrm{symmetric}\mathrm{on}A. \end{gathered}$$

(ii) R is reflexive and S is any relation.
$$\begin{gathered} \mathrm{Suppose}a\in A.\mathrm{Then}, \\ \left(a,a\right)\in R\left[\mathrm{Since}R\mathrm{is}\mathrm{reflexive}\right] \\ \Rightarrow \left(a,a\right)\in R\cup S \\ \Rightarrow R\cup S\mathrm{is}\mathrm{reflexive}\mathrm{on}A. \end{gathered}$$