If A ={1,2,3,4}, define relations on A which have properties of being
(i) Reflexive, transitive but not symmetric.

(ii) symmetric but neither reflexive nor transitive.

(iii) reflexive,symmetric and transitive.

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Solution

(i) Given that, A={1,2,3,4}
Let ${R_{1} = {(1, 1), (1, 2), (2, 3), (2, 2), (1, 3), (3, 3)}$

$R_{1}$ is reflexive, since, (1,1)(2,2)(3,3) lie in $R_{1} .$
Now. $(1, 2) \in R_{1}, (2, 3) \in R_{1} \Rightarrow (1, 3) \in R_{1}$
Hence, $R_{1}$ is also transitive but $(1, 2) \in R_{1} w h e r e a s (2, 1) / \in R_{1} .$
So, it is not symmetric.

(ii) Given that, A={1,2,3,4}
Let $R_{2} = {(1, 2), (2, 1)}$
Now, $(1, 2) \in R_{2} . (2, 1) \in R_{2}$

So, it is symmetric. Since (,1) is not an element, it is neither reflexive of transitive.

(iii) Given that, A={1,2,3,4}
Let $R_{3} = {(1, 2), (2, 1), (1, 1), (2, 2), (3, 3), (1, 3) (3, 1), (2, 3)}$
Hence, $R_{3}$ is reflexive, symmetric and transitive.

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Similar questions

A = {1, 2, 3, 4}

A = {1, 2, 3, 4}

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

Given an example of a relation. Which is

(i) Symmetric but neither reflexive nor transitive.

(ii) Transitive but neither reflexive nor symmetric.

(iii) Reflexive and symmetric but not transitive.

(iv) Reflexive and transitive but not symmetric.

(v) Symmetric and transitive but not reflexive.

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