On the set `N` of all natural numbers, a relation `R` is defined as follows: `n R m ` <=> Each of the natural numbers `n` and `m` leaves the same remainder less than 5 when divided by 5. Show that `R` is an equivalence relation. Also, obtain the pairwise disjoint subsets determined by `R` .

Explanation

`R` is reflexive since for each `a \in N, a R a`. `R` is symmetric since if `a R b`, then bRa for `a, b` `\in N`. Also, `R` is transitive since for `a, b, c \in N`, if aRb and `b R c`, then aRc. Hence `R` is an equivalence relation in `N` which will partition the set `N` into the pairwise disjoint subsets. The equivalent classes are as mentioned below:



$$ A_{0}=\{5,10,15,20 \ldots\} $$
$$ A_{1}=\{1,6,11,16,21 \ldots\} $$
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