Question

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

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

Answer 1

Answer: (B)

$R_1=\{(a,b)\in N\times N:|a-b|\le 13\}$
$R_2=\{(a,b)\in N\times N:|a-b|\ne 13\}$
For $R_1$ :
i) Reflexive relation
$(a,a)\in N\times N:|a-a|\le 13$
ii) Symmetric relation
$(a,b)\in R_1,(b,a)\in R_1:|b-a|\le 13$
iii) Transitive relation
$(a,b)\in R_1,(b,c)\in R_1,(a,c)\in R_1:$
$(1,3)\in R_{1,}(3,16)\in R_1\text{ but }(1,16)\notin R_1$
For $R_2$ :
i) Reflexive relation
$(a,a)\in N\times N:|a-a|\ne 13$
ii) Symmetric relation
$(b,a)\in N\times N:|b-a|\ne 13$
iii) Transitive relation
$(a,b)\in R_2,(b,c)\in R_2,(a,c)\in R_2$
$(1,3)\in R_{2,}(3,14)\in R_2\text{ but }(1,14)\notin R_2$