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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
Solution
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.
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