Question

(x,y)belong r x-y is divisible by n is an equvialence relation onz

Answer 1

hey mate ur ans
A = { x ∈ Z : 0 ≤ x ≤ 12} = {0,1,2,3,4,5,6,7,8,9,10,11,12}
(i) R = { (a,b) : |a - b| is a multiple of 4}
For any element a ∈A, we have (a, a) ∈ R as |a - a = 0|is a multiple of 4.
∴R is reflexive.
Now, let (a, b) ∈ R ⇒ |a - b| is a multiple of 4.
⇒ |-(a - b)| = ⇒ |b - a| is a multiple of 4.
⇒ (b, a) ∈ R
∴R is symmetric.
Now, let (a, b), (b, c) ∈ R.
⇒ |(a - b)| is a multiple of 4 and |(b - c)| is a multiple of 4. 
⇒ (a - b) is a multiple of 4 and (b - c) is a multiple of 4.
⇒ (a - c) = (a – b) + (b – c) is a multiple of 4. 
⇒ |a - c| is a multiple of 4.
⇒ (a, c) ∈R
∴ R is transitive.
Hence, R is an equivalence relation.
The set of elements related to 1 is {1, 5, 9} since