# (i) (a) A function ƒ : X → Y is onto if range of  ƒ = …………. (b) Let ƒ : {1, 3, 4} {3, 4, 5} and g: {3, 4, 5} → {6, 8, 10} be functions defined

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(i) (a) A function ƒ : X → Y is onto if range of

ƒ = ………….

(b) Let ƒ : {1, 3, 4} {3, 4, 5} and g: {3, 4, 5} → {6, 8, 10} be functions defined

by

ƒ (1) = 3, ƒ (3) = 4, ƒ (4) = 5;

g (3) = 6, g(4) = 8, g(5) = 8 ,then (goƒ) (3) = …………..

(ii) Let Q be the set of Rational numbers and ‘*’ be the binary operation on Q defined by a *b = $\frac{ab}{4}$ for all a,b in Q

(a) What is the identity element of ‘ * ’on Q?

(b) Find the inverse element of * ’ on Q.

(c) Show that a * (b * c) = (a * b) * c, ∀a,b,c ∈ Q.

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(i) (a) Range of f = y

(b) (g of) (3) = g(gf(3)) = g(4) = 8

(ii) (a) a*e = a ⇒ $\frac{ae}{4}$ = a ⇒ e = 4

e*a = a ⇒ $\frac{ea}{4}$ = a ⇒ e = 4

identity element = 4

(b) a*b = e ⇒ $\frac{ab}{4}$ = 4 ⇒ b = $\frac{16}{a}$

b*a = e ⇒ $\frac{ba}{4}$ = e ⇒ $\frac{ba}{4}$ = 4 ⇒ b = $\frac{16}{a}$

inverse element = b = $\frac{16}{a}$

(c) a*(b*c) = a*$(\frac{bc}{4})$ = $\frac{a \frac{bc}{4}}{4}$ = $\frac{abc}{16}$

(a*b)*c = $\frac{ab}{4}$*c = $\frac{\frac{ab}{4}c}{4}$ = $\frac{abc}{16}$

Hence ; a*(b*c) = (a*b)*c