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1.Relation and Function
medium
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
A
Only $R_1$ is an equivalence relation
B
Only $R_2$ is an equivalence relation
C
$R_1$ and $R_2$ both are equivalence relations
D
Neither $R_1$ nor $R_2$ is an equivalence relation
(JEE MAIN-2024)
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.
Standard 12
Mathematics
