# (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 by

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(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 by

f(x) = $\frac{x}{x+2}$ is  one-one.

(iii) Let ‘*’ be a binary operation on Q dened by a*b = a*b = $\frac{ab}{6}$ ’.Find the inverse of 9 with respect to ’ * ’.

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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