Question

In the set of natural numbers N, define a relation R as follows: ∀ n, m ∈ N, nRm if on division by 5 each of the integers n and m leaves the remainder less than 5, i.e. one of the numbers 0, 1, 2, 3 and 4. Show that R is equivalence relation. Also, obtain the pairwise disjoint subsets determined by R

Answer 1

R is reflexive since for each a ∈ N, aRa.

R is symmetric since if aRb, then bRa for a, b ∈ N.

Also, R is transitive since for a, b, c ∈ N, if aRb and bRc, 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:

A0 = {5, 10, 15, 20 ...}

A1 = {1, 6, 11, 16, 21 ...}

A2 = {2, 7, 12, 17, 22, ...}

A3 = {3, 8, 13, 18, 23, ...}

A4 = {4, 9, 14, 19, 24, ...}

It is evident that the above five sets are pairwise disjoint and

A₀ ∪ A₁ ∪ A₂ ∪ A₃ ∪ A₄ = `∪_("i" = 0)^4  "A"_"i"` = N.