Question

Let $ R $ and $ S $ be any two equivalence relations on a set $ X $ . Then which of the following is incorrect statement

• A. $ R \cup S $ is an equivalence relation on $ X $
• B. $ R^{-1} $ is an equivalence relation on $ X $
• C. $ R^{- 1} \cap S^{-1} $ is an equivalence relation on $ X $
• D. $ \Delta $ is an equivalence relation on $ X $ , where $ \Delta $ is the diagonal relation on $ X $ .

Answer 1

Answer: (A)

We know that the union of two equivalence relation on a set is not necessarily an equivalence relation on the set.
For example let $x=\left\{a, b, c\right\}$
$R=\left\{\left(a,a\right), \left(b,b\right)\left(c,c\right), \left(a, b\right), \left(b, a\right)\right\}$
$S=\left\{\left(a,a\right), \left(b,b\right)\left(c,c\right), \left(b,c\right), \left(c,b\right)\right\}$
Clearly, $R$ and $S$ are equivalence relation
But $R \cup S$ is not transitive because $\left(a, b\right)\in R \cup S$
and $\left(b,c\right)\in R \cup S$ but $\left(a, c\right) \notin R \cup S $
Hence, $R \cup S$ is not equivalence relation