If $A = {1, 2, 3, 4}$ define relations on A which have properties of being symmetric but neither reflexive nor transitive.

R_{2} = {(1, 2), (2, 1)}

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R_{2} = {(1, 1), (2, 1)}

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R_{2} = {(1, 2), (2, 1), (2, 3)}

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none of these

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Solution

The correct option is A $R_{2} = {(1, 2), (2, 1)}$
Consider option A,

$R_{2} = {(1, 2), (2, 1)}$

Test for symmetry: A relation $R$ in $A$ is said to be symmetric, if $(a_{1}, a_{2}) \in R ⟹ (a_{2}, a_{1}) \in R$ for $a_{1}, a_{2} \in A$ .

$(1, 2) \in R_{2} ⟹ (2, 1) \in R_{2}$

Hence $R_{2} = {(1, 2), (2, 1)}$ is symmetric.

Test for reflexive : A relation $R$ in $A$ is said to be reflexive, if $(a, a) \in R$ for every $a \in A$ .

Since $(1, 1), (2, 2), (3, 3), (4, 4) \notin R_{2}$ , $R_{2}$ is not reflexive.

Test for transitive: A relation $R$ in $A$ is said to be transitive, if $(a_{1}, a_{2}) \in R$ and $(a_{2}, a_{3}) \in R ⟹ (a_{1}, a_{3}) \in R$ for all $a_{1}, a_{2}, a_{3} \in A$ .

$(1, 2) \in R_{2}$ and $(2, 1) \in R_{2}$ but $(1, 1) \notin R_{2}$ Hence $R_{2}$ is not transitive.

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