Let R_1 and R_2 be two relations on a set A. | vedclass

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(D) $1$. Transitivity: If $R_1$ and $R_2$ are transitive,then $R_1 \cap R_2$ is transitive. This is a true statement.$2$. Reflexivity: If $R_1$ and $R

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If $R_1$ and $R_2$ are equivalence,then $R_1 \cup R_2$ is also equivalence.

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(D) $1$. Transitivity: If $R_1$ and $R_2$ are transitive,then $R_1 \cap R_2$ is transitive. This is a true statement.$2$. Reflexivity: If $R_1$ and $R_2$ are reflexive,then $R_1 \cup R_2$ is reflexive. This is a true statement.$3$. Symmetry: If $R_1$ and $R_2$ are symmetric,then $R_1 \cap R_2$ is symmetric. This is a true statement.$4$. Equivalence: If $R_1$ and $R_2$ are equivalence relations,their union $R_1 \cup R_2$ is not necessarily an equivalence relation because it may fail to be transitive. For example,let $A = \{1, 2, 3\}$. Let $R_1 = \{(1, 1), (2, 2), (3, 3), (1, 2), (2, 1)\}$ and $R_2 = \{(1, 1), (2, 2), (3, 3), (2, 3), (3, 2)\}$. Both are equivalence relations. However,$R_1 \cup R_2 = \{(1, 1), (2, 2), (3, 3), (1, 2), (2, 1), (2, 3), (3, 2)\}$. Here,$(1, 2) \in R_1 \cup R_2$ and $(2, 3) \in R_1 \cup R_2$,but $(1, 3) 
otin R_1 \cup R_2$. Thus,it is not transitive. Therefore,the statement in option $D$ is incorrect.

Let $R_1$ and $R_2$ be two relations on a set $A$. Choose the incorrect statement.

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