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.
