Let X=R R.

Question

Let $X=R	imes R$. Define a relation R on X as :

$({a}_{1},{b}_{1})R({a}_{2},{b}_{2})\Leftrightarrow {b}_{1}={b}_{2}$

Statement I: $\, R$ is an equivalence relation.

Statement II : For some $(a,b)\in X$, the $setS={(x,y)\in X:(x,y)R(a,b)}$ represents a line parallel to $y=x$.

In the light of the above statements, choose the correct answer from the options given below :

Accepted Answer

Statement - I :

Reflexive : $\left(a_1, b_1\right) R\left(a_1, b_1\right) \Rightarrow b_1=b_1 \quad$ True

$\left.\begin{gathered} \text { Symmetric : } \quad  \left(\mathrm{a}_1, \mathrm{~b}_1\right) \mathrm{R}\left(\mathrm{a}_2, \mathrm{~b}_2\right) \Rightarrow \mathrm{b}_1=\mathrm{b}_2 \ \left(\mathrm{a}_2, \mathrm{~b}_2\right) \mathrm{R}\left(\mathrm{a}_1, \mathrm{~b}_1\right) \Rightarrow \mathrm{b}_2=\mathrm{b}_1 \end{gathered}\right\}$ True

$\left.\begin{array}{ll} \text { Transitive: } & \left(a_1, b_1\right) R\left(a_2, b_2\right) \Rightarrow b_1=b_2 \ &\left(a_2, b_2\right) R\left(a_3, b_3\right) b_2=b_3 \end{array}\right\} b_1=b_3$

$$\Rightarrow\left(\mathrm{a}_1, \mathrm{~b}_1\right) \mathrm{R}\left(\mathrm{a}_3 \cdot \mathrm{~b}_3\right) \Rightarrow \text { True }$$

Hence Relation $R$ is an equivence relation Statement-I is true.

For statement - II $\Rightarrow \mathrm{y}=\mathrm{b}$ so False

Solutions · 1