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 $(a, b) R_2(c, d) \Leftrightarrow \mathrm{a}+ \mathrm{d}=\mathrm{b}+\mathrm{c}$ for all $(\mathrm{a}, \mathrm{b}),(\mathrm{c}, \mathrm{d}) \in \mathrm{N} \times \mathrm{N}$. Then
Select the correct option:
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
✓ Correct! Well done.
✗ Incorrect. Try again or view the solution.
Solution
$$
\begin{aligned}
& a R_1 b \Leftrightarrow a^2+b^2=1 ; a, b \in R \\
& (a, b) R_2(c, d) \Leftrightarrow a+d=b+c ;(a, b),(c, d) \in N
\end{aligned}
$$
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.
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