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Consider the relations $R_{1}$ and $R_{2}$ defined as a $a R_{1} b \Leftrightarrow a^{2} + b^{2} = 1$ for all $a, b \in R$ and $(a, b) R_{2} (c, d) \Leftrightarrow a + d = b + c$ for all $(a, b), (c, d) \in N \times N$ . Then

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Only $R_{1}$ is an equivalence relation

$R_{2}$ Only is an equivalence relation

$R_{1}$ and $R_{2}$ both are equivalence relations

Neither $R_{1}$ nor $R_{2}$ is an equivalence relation

answer is B.

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Detailed Solution

For $R_{1}$ : Not reflexive symmetric not transitive
for $R_{2}$ : is reflexive, symmetric and transitive Hence only $R_{2}$ is equivalence relation.

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