Question

(i) Let R be a relation defined on A{1,2,3} by R = {(13),(3,1),(2,2)} is

(a) Reflexive

(b) Symmetric

(C) Transitive

(d) Reflexive but not transitive.

(ii) Find fog and gof if ƒ(x) = |x+1| and g(x) = 2x – 1

(iii) Let * be a binary operation defined on N x N by (a,b) * (c,d.) = (a + c, b + d)
Find the identity element for * if it exists.

Answer 1

(i) (b) Symmetric

(ii) ƒog(x) = |g(x) + 1| = |2x – 1 + 1| = |2x|
goƒ(x) = 2 ƒ(x) – 1 = 2 |x + 1| – 1

(iii) Let e =(e₁, e₂) be the identity element of the operation in ? N x N then, (a,b)*(e₁, e₂) = (a + e₁, b + e₂) ≠ (a,b) Since, e₁ ≠ 0, e₂ ≠ 0

Therefore identity element does not exists.