Question

(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.

Answer 1

(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