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Question
In the set Z of integers, define mRn if m − n is divisible by 7. Prove that R is an equivalence relation
Solution
Z = set of all integers
Relation R is defined on Z by m R n if m – n is divisible by 7.
R = {(m, n), m, n ∈ Z/m – n divisible by 7}
m – n divisible by 7
∴ m – n = 7k where k is an integer.
a) Reflexive:
m – m = 0 = 0 × 7
m – m is divisible by 7
∴ (m, m) ∈ R for all m ∈ Z
Hence R is reflexive.
b) Symmetric:
Let (m, n) ∈ R ⇒ m – n is divisible by 7
m – n = 7k
n – m = – 7k
n – m = (– k)7
∴ n – m is divisible by 7
∴ (n, m) ∈ R.
c) Transitive:
Let (m, n) and (n, r) ∈ R
m – n is divisible by 7
m – n = 7k ......(1)
n – r is divisible by 7
n – r = 7k1 ......(2)
(m – n) + (n – r) = 7k + 7k1
m – r = (k + k1) 7
m – r is divisible by 7.
∴ (m, r) ∈ R
Hence R is transitive.
R is an equivalence relation.
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