Question

If $R$ is the real line.

Consider the following subsets of the plane $R\times R;S=\{(x,y):y=x+1;0<x<2\};T=\{(x,y):x-y\text{is an integer}\}$ Which one of the following is correct?

• A. $T$ is an equivalence relation on $R$ but $S$ is not
• B. Neither $S$ nor $T$ is an equivalence relation on $R$
• C. Both $S$ and $T$ are equivalence relations on $R$
• D. $S$ is an equivalence relation on $R$ but $T$ is not

Answer 1

The correct option is A

$T$ is an equivalence relation on $R$ but $S$ is not

Explanation for the correct option:

Step 1: Check whether the given relation $T=\{(x,y):x-y\text{is an integer}\}$ is an equivalence relation or not.

If a relation is reflexive, symmetric, and transitive, it is said to be an equivalence relation.

Reflexive: A relation is reflexive, if $\left(a,a\right)\in R$, for every $a\in R$.

Symmetric: A relation is symmetric, if $\left(a,b\right)\in R$, then $\left(b,a\right)\in R$.

Transitive: A relation is transitive if $\left(a,b\right)\in R$ and $\left(b,c\right)\in R$, then $\left(a,c\right)\in R$.

• Let $x\in R$, then $x-x=0$ is an integer,

Which implies if function $(x,x)\in T$, which shows $T$ is reflexive;

• Let $(x,y)\in R$ , if $x-y$ is an integer, then $y-x$ is also an integer, which implies $(y,x)\in R$ , then

The function $T$ is symmetric relation.

• Assume $(x,y,z)\in R$
• Let $(x,y)\in T$ and $(y,z)\in T$ then $x-y\in I_{\mathit{1}}$ is an integer and $y-z\in I_2$ is also an integer

Then add both the above relation, then

$\begin{array}{rcl}{I}_1+I_2& =& x-y+y-z\\ & =& x-z\in I,\text{an integer}\end{array}$

This implies $(x,z)\in T$, which shows that $T$ is transitive.

Therefore, the relation $T$ is an equivalence relation, because $T$ is reflexive, symmetric, and transitive.

Step 2: Check whether the given relation $S=\{(x,y):y=x+1;0<x<2\}$ is an equivalence relation or not.

If $x\in R$, assume $(x,x)\in S$, then

$x=x+1$ is not possible if $0<x<2$,

Therefore relation $S$ is not reflexive which implies it is not an equivalence relation.

Hence, the correct option is (A).