Question:
Let n be a fixed positive integer. Define a relation R in Z as follows: ∀ a, b ∈ Z, aRb if and only if a – b is divisible by n. Show that R is an equivalance relation.
Solution:
Given ∀ a, b ∈ Z, aRb if and only if a – b is divisible by n.
Now, for
aRa ⇒ (a – a) is divisible by n, which is true for any integer a as ‘0’ is divisible by n.
Thus, R is reflective.
Now, aRb
So, (a – b) is divisible by n.
⇒ – (b – a) is divisible by n.
⇒ (b – a) is divisible by n
⇒ bRa
Thus, R is symmetric.
Let aRb and bRc
Then, (a – b) is divisible by n and (b – c) is divisible by n.
So, (a – b) + (b – c) is divisible by n.
⇒ (a – c) is divisible by n.
⇒ aRc
Thus, R is transitive.
So, R is an equivalence relation.