Question

Let R and S be two relations on a set A. Then which of the following is incorrect?

• A. R and S are transitive, then R $\cup$ S is also transitive.
• B. R and S are transitive, then R $\cap$ S is also transitive.
• C. R and S are reflexive, then R $\cap$ S is also reflexive.
• D. R and S are symmetric, then R $\cup$ S is also symmetric

Answer 1

Answer: (A)

(i) Transitivity property usually does not hold in unions.
Consider the set A = {1, 2, 3} and
R = {(1, 2)} and S = {(2, 3)}.
Clearly R and S are transitive but R $\cup$ S = {(1, 2), (2, 3)} is not transitive.
$\therefore$ False.
(ii) Let (a, b), (b, c) $\in$ R $\cap$ S
$\Rightarrow (a,b),(b,c)\in R$ and $(a,b)(b,c)\in S$
$\Rightarrow (a,c)\in R,(a,c)\in S$
$\Rightarrow (a,c)\in R\cap S.$
$\therefore$ If R, S are transitive then R $\cap$ S is also transitive
(iii & iv)
R, S are reflexive, so $(a,a)\in R,(a,a)\in$
$S\forall a\in A$
$\therefore \forall a\in A,(a,a)\in R\cap S$ and $R\cup S$ and so $R\cup S$ and $R\cap S$ both reflexive.