Given an example of a relation. Which is
(iv) Reflexive and transitive but not symmetric.

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Solution

Define a relation R in R(real numbers) as:
$R = {(a, b) : a^{3} \geq b^{3}}$
Clearly $(a, a) a s a^{3} = a^{3}$ . Therefore, R is reflexive.
Now, $(2, 1) \in R (a s 2^{3} \geq 1^{3})$ But, $(1, 2) / \in R$ (as $1^{3} < 2^{3})$
Therefore, R is not symmetric .
Now, let $(a, b), (b, c) \in R$
$\Rightarrow a^{3} \geq b^{3}$ and $b^{3} \geq c^{3} \Rightarrow a^{3} \geq c^{3} \Rightarrow (a, c) \in R$ . Therefore, R is transitive.
Hence, relation R is reflexive and transitive but not symmetric.

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