Question

Let $R _{1}=\{( a , b ) \in N \times N :| a - b | \leq 13\}$ and
$R _{2}=\{( a , b ) \in N \times N :| a - b | \neq 13\} .$ 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 | \leq 13\}$
$ R _{2}=\{( a , b ) \in N \times N :| a - b | \neq 13\}$
For $R _{1}$ :
i) Reflexive relation
$(a, a) \in N \times N:|a-a| \leq 13$
ii) Symmetric relation
$( a , b ) \in R _{1},( b , a ) \in R _{1}:| b - a | \leq 13$
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
$(a, a) \in N \times N:|a-a| \neq 13$
ii) Symmetric relation
$(b, a) \in N \times N:|b-a| \neq 13$
iii) Transitive relation
$( a , b ) \in R _{2},( b , c ) \in R _{2},( a , c ) \in R _{2}$
$(1,3) \in R _{2,}(3,14) \in R _{2} \text { but }(1,14) \notin R _{2}$