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Let $R _{1}=\{( a , b ) \in N \times N 😐 a - b | \leq 13\}$ and $R _{2}=\{( a , b ) \in N \times N 😐 a - b | \neq 13\} .$ Thenon $N$
Both $R_{1}$ and $R_{2}$ are equivalence relations
Neither $R_{1}$ nor $R_{2}$ is an equivalence relation.
$R_{1}$ is an equivalence relation but $R_{2}$ is not
$R_{2}$ is an equivalence relation but $R_{1}$ is not
Solution
$R_{1}=\{(a, b) \in N \times N:|a-b| \leq 13\}$
$R_{2}=\{(a, b) \in N \times N:|a-b| \neq 13\}$.
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 $R _{2}$ :
$(i) \,Reflexive\,\, relation$
$(a, a) \in N \times N:|a-a| \neq 13$
$(ii)\, Symmetric\,\, relation$
$(b, a) \in N \times N:|b-a| \neq 13$
$(iii)\, Transitive \,\,relation$
$( a , b ) \in R _{2},( b , c ) \in R _{2},( a , c ) \in R _{2}$
$(1,3) \in R _{2,}(3,14) \in R _{2} \text { but }(1,14) \notin R _{2}$
