(i) Give a relation on a set A = {1,2,3,4} which is reexive , symmetr

1.	(i) Give a relation on a set A = {1,2,3,4} which is reexive , symmetric and not transitive. (ii) Show that ƒ : [-1,1] → R given byf(x) = \(\frac{x}{x+2}\) is one-one.(iii) Let ‘’ be a binary operation on Q dened by ab = ab = \(\frac{ab}{6}\) ’.Find the inverse of 9 with respect to ’ ’.
Answer» R = {(1,1)(2,2),(3,3),(4,4),(1,2),(2,1),(1,3),(3,1)} f(x₁) = f(x₂) ⇒ \(\frac{x_1}{x_1+2}\) =\(\frac{x_2}{x_2 + 2}\) ⇒ x₁(x₂+ 2) = x₂ (x₁ + 2) ⇒ x₁x₂ + 2x₁ = x₂x₁ + 2x₂ ⇒2x₁ = 2x₂ ⇒ x₁ = x₂ Hence one-one (iii) ae = a⇒ \(\frac{ae}{6}\) = a ⇒ e=6 ea = a⇒ \(\frac{ea}{6}\) = a ⇒ e = 6 identity element = 6 ab = e ⇒ 9b = 6 ⇒ \(\frac{9b}{6}\) = 6 ⇒ b = \(\frac{36}{9}\) = 4 ba = e ⇒ b9 = 6 ⇒ \(\frac{b9}{6}\) = 6 ⇒ b = \(\frac{36}{9}\) = 4 Inverse element = b = 4

(i) Give a relation on a set A = {1,2,3,4} which is reexive , symmetric and not transitive. (ii) Show that ƒ : [-1,1] → R given byf(x) = \(\frac{x}{x+2}\) is one-one.(iii) Let ‘’ be a binary operation on Q dened by ab = ab = \(\frac{ab}{6}\) ’.Find the inverse of 9 with respect to ’ ’.

Answer»

R = {(1,1)(2,2),(3,3),(4,4),(1,2),(2,1),(1,3),(3,1)}

f(x₁) = f(x₂) ⇒ \(\frac{x_1}{x_1+2}\) =\(\frac{x_2}{x_2 + 2}\)

⇒ x₁(x₂+ 2) = x₂ (x₁ + 2)

⇒ x₁x₂ + 2x₁ = x₂x₁ + 2x₂

⇒2x₁ = 2x₂ ⇒ x₁ = x₂

Hence one-one

(iii) a*e = a⇒ \(\frac{ae}{6}\) = a ⇒ e=6

e*a = a⇒ \(\frac{ea}{6}\) = a ⇒ e = 6

identity element = 6

a*b = e ⇒ 9*b = 6 ⇒ \(\frac{9b}{6}\) = 6 ⇒ b = \(\frac{36}{9}\) = 4

b*a = e ⇒ b*9 = 6 ⇒ \(\frac{b9}{6}\) = 6 ⇒ b = \(\frac{36}{9}\) = 4

Inverse element = b = 4