### Sets And Relations Question 3

#### Question 3 - 2024 (01 Feb Shift 2)

Consider the relations $R _1$ and $R _2$ defined as $a R _1 b \Leftrightarrow a^{2}+b^{2}=1$ 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

(1) Only $R _1$ is an equivalence relation

(2) Only $R _2$ is an equivalence relation

(3) $R _1$ and $R _2$ both are equivalence relations

(4) Neither $R _1$ nor $R _2$ is an equivalence relation

## Show Answer

#### Answer (2)

#### Solution

$a R _1 b \Leftrightarrow a^{2}+b^{2}=1 ; a, b \in R$

$(a, b) R _2(c, d) \Leftrightarrow a+d=b+c ;(a, b),(c, d) \in N$

for $R _1$ : Not reflexive symmetric not transitive

for $R _2: R _2$ is reflexive, symmetric and transitive Hence only $R _2$ is equivalence relation.