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

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Let R_1 and R_2 be symmetric relations.


Let us prove that R_1\cap R_2 is also symmetric relation. 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, we conclude that (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\cap R_2 is also symmetric.


Let us prove that R_1\cup R_2 is also symmetric relation. 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, we conclude that (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.


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