If R and S are two non-empty relations on a s | vedclass

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Question

(A) Let $A = \{1, 2, 3\}$. Consider $R = \{(1, 2)\}$ and $S = \{(2, 3)\}$. Both $R$ and $S$ are transitive relations on $A$ because they do not contai

Vedclass · Accepted Answer

$R$ and $S$ are transitive $\implies R \cup S$ is transitive

Vedclass · Answer

(A) Let $A = \{1, 2, 3\}$. Consider $R = \{(1, 2)\}$ and $S = \{(2, 3)\}$. Both $R$ and $S$ are transitive relations on $A$ because they do not contain any pairs $(a, b)$ and $(b, c)$ such that $(a, c)$ is missing.Now,$R \cup S = \{(1, 2), (2, 3)\}$.For $R \cup S$ to be transitive,since $(1, 2) \in R \cup S$ and $(2, 3) \in R \cup S$,we must have $(1, 3) \in R \cup S$. However,$(1, 3) 
otin R \cup S$.Thus,$R \cup S$ is not necessarily transitive. Therefore,the statement '$R$ and $S$ are transitive $\implies R \cup S$ is transitive' is false.

If $R$ and $S$ are two non-empty relations on a set $A$,then which of the following statements is false?

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