Mathematical Foundation of Computer Science

DHARM

22 MATHEMATICAL FOUNDATION OF COMPUTER SCIENCE

1.17 Which of the following graphs represents injective, surjective, and bijective function?

(i) Circle (ii) Cissoid (iii) Straight line

y

×

y

×

y

y

x

(iv) Parabola (v) Hyperbola

1.18 Let a function f : R → R be defined i.e., f(x) = 1 if x is rational and f(x) = – 1 if x is irrational then

find

(i)f(2) (ii)f(1/2)

(iii)f(1/3) (iv)f(22/7)

(v)f( 2 )(vi)f(^48 )

1.19 Let a function f : R → R be defined as

f(x) =

12 2 4

47

312

2

2

−−≤≤

+≤≤

−≤<

R

S

|

T

|

xx

xx

xx

for

for 5

for 8

Find :

(i)f(– 3) (ii)f(4)

(iii)f(7) (iv)f(12)

(v)f(u – 2)

1.20 Let f be a function such that f–1 exist then state properties of the function f.

1.21 Let a function f : R → R be defined as f(x) = x^2 , then find

(i)f–1(– 4) (ii)f–1(25)

(iii)f–1(4 ≤ x ≤ 25) (iv)f–1(– ∞ < x ≤ 0)

1.22 Find the f–1 if function f(x) = (4x^2 – 9)/(2x +3) (by assuming f is surjective).

1.23 Let function f and g are surjective then show that

(g o f)–1 = (f–1 o g–^1 )

1.24 Find x and y in the following ordered pair :

(i)(x + 3, y – 4) = (3, 5) (ii)(x–2, y + 3) = (y + 4, 2x – 1)

1.25 Let X = {1, 2, 3, 4} and Y = {a, b, c}, can a injective and surjective function

f : X → Y may be defined. Give reasons.