Question

Let $R_1=\left\{\right(a,b)\in N\times N:|a-b|\le 13\}$ and $R_2=\left\{\right(a,b)\in N\times N:|a-b|\ne 13.\}$ Then on $N:$

• A. $R_1$ is an equivalence relation but $R_2$ is not
• B. Neither $R_1$ nor $R_2$ is an equivalence relation
• C. Both $R_1$ and $R_2$ are equivalence relations
• D. $R_2$ is an equivalence relation but $R_1$ is not

Answer 1

The correct option is B Neither $R_1$ nor $R_2$ is an equivalence relation

$R_1=\left\{\right(a,b)\in N\times N:|a-b|\le 13.\}\text{and:}$
$R_2=\left\{\right(a,b)\in N\times N:|a-b|\ne 13\}$
In $R_1:|2-11|=9\le 13$

$\therefore (2,11)\in R_1\text{and}(11,19)\in R_1\text{but}$
$(2,19)\notin R_1$
$\therefore$ $R_1$ is not transitive
Hence $R_1$ not equivalence
In $R_2:(13,3)\in R_2\text{and}(3,26)\in R_2\text{but}$
$(13,26)\notin R_2(\because |13-26|=13)$
$\therefore R_2$ is not transitive
Hence $R_2$ is not equivalence.