For α ∈ N , consider a relation R on N given by R = x , y : 3 x + α y is a multiple of 7 . The relation R is an equivalence relation if and only if

Step-by-step solution

Given, a relation R on N given by R = x , y : 3 x + α y is a multiple of 7 , Now for R to be reflexive ⇒ x R x ⇒ 3 x + α x = 7 x ⇒ 3 + α x = 7 K ⇒ 3 + α = 7 λ ⇒ α = 7 λ - 3 = 7   N + 4 ,   where  K , λ , N ∈ I So, when α divided by 7 , remainder is 4 . Now R to be symmetric x R y ⇒ y R x 3 x + α y = 7 N 1 , 3 y + α x = 7 N 2 ⇒ 3 + α x + y = 7 N 1 + N 2 = 7 N 3 Which holds when 3 + α is mul