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Question
Let n be a fixed positive integer. Define a relation R on Z as follows:
(a, b) ∈ R ⇔ a − b is divisible by n.
Show that R is an equivalence relation on Z.
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Solution
We observe the following properties of R. Then,
Reflexivity :
Let a ∈ N
Here,
a − a = 0 = 0 × n
⇒ a−a is divisible by n
⇒ (a, a) ∈ R
⇒ (a, a) ∈ R for all a ∈ Z
So, R is reflexive on Z.
Symmetry :
Let (a, b) ∈ R
Here,
a−b is divisible by n
⇒ a−b = np for some p ∈ Z
⇒ b−a = n (−p)
⇒ b−a is divisible by n [ p ∈ Z⇒ − p ∈ Z]
⇒ (b, a) ∈ R
So, R is symmetric on Z.
Transitivity :
Let (a, b) and (b, c) ∈ R
Here, a−b is divisible by n and b−c is divisible by n.
⇒ a−b= np for some p ∈ Z
and b−c = nq for some q ∈ Z
a−b+ b−c = np + nq
⇒ a−c = n (p+q)
⇒ (a, c)∈ R for all a, c ∈ Z
So, R is transitive on Z.
Hence, R is an equivalence relation on Z.
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