Show that each of the relation R in the set A = {x ∈ Z : 0 ≤ x ≤ 12}, given by (i) R = {(a, b) : |a – b| is a multiple of 4} (ii) R = {(a, b) : a = b}

Answer
(i) R = {(a, b) : |a – b| is a multiple of 4}
 A = {x ∈ Z : 0 ≤ x ≤ 12}
∴ A = {0,1,2,3,4,5,6,7,8,9,10,11,12}
(i)
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| = 4k , k is any integer
⇒ (b, a) = |b-a| = |-4k| = 4k   it is also multiple of 4 ∈ R
∴R is symmetric.
Now, let (a, b), (b, c) ∈ R.
∴|a-b| = 4k  and |b - c|  = 4s    , here k and s are integers 
⇒ (a, c) = |a-b +b - c | = |4k + 4 s| = 4|k+s|
hence it is also 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



 (ii) R = {(a, b): a = b}
For any element a ∈A, we have (a, a) ∈ R, since a = a.
∴R is reflexive.
Now, let (a, b) ∈ R.
⇒ a = b
⇒ b = a
⇒ (b, a) ∈ R
∴R is symmetric.
Now, let (a, b) ∈ R and (b, c) ∈ R.
⇒ a = b and b = c
⇒ a = c
⇒ (a, c) ∈ R
∴ R is transitive.
Hence, R is an equivalence relation.
The elements in R that are related to 1 will be those elements from set A which are equal

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Show that each of the relation R in the set A = {x ∈ Z : 0 ≤ x ≤ 12}, given by
(i) R = {(a, b) : |a – b| is a multiple of 4}
(ii) R = {(a, b) : a = b}  in below box