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Q.
Let R and S be two relations on a set A. Then which of the following is incorrect?
Relations and Functions - Part 2
Solution:
(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.