Let $X=\mathbf R imes\mathbf R$. 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 set $S=\{(x,y)\in X:(x,y)R(a,b)\}$ represents a line parallel to $y=x$. In the light of the above statements, choose the correct answer from the options given below :

Question

Let $X=\mathbf R	imes\mathbf R$. 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 set $S=\{(x,y)\in X:(x,y)R(a,b)\}$ represents a line parallel to $y=x$. In the light of the above statements, choose the correct answer from the options given below :

Dr. Abubacker · Accepted Answer

[Solution by Dr. Abubacker] $b_1=b_2$ is reflexive, symmetric and transitive, so $R$ is an equivalence relation. $S=\{(x,y):y=b\}$ is always a horizontal line (parallel to the $x$-axis), never parallel to $y=x$, so Statement II is false. Answer (2).

Explanation

Reviewed by Dr. Abubacker

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