Q1 If the relation R in the set A, where A = {1, 2, 3, 4, 5, 6}, is defined by R = {(x, y) : y is divisible by x}, then express R in the roster form. Also determine whether the relation R is
(i) reflexive
(ii) symmetric
(iii) transitive

Solution

Q2 If R is the relation defined on the set of natural numbers N as follows:
R = {(x, y); x, y ∈ N, 2x + y = 41}, find the domain and the range of the relation R.
Determine whether the relation is reflexive, symmetric and transitive.

Solution

Q3 Determine whether each of the following relation are reflexive , symmetric and transitive
(i) R = {(a, b) : a, b ∈ N, a < b}
(ii) R = {(a, b) : a, b ∈ R, a ≤ b }

Solution

Q4 Show that the relation R in the set A = {x ∈ W , 0≤x≤17} given by
R={(a,b) : |a — b| is a multiple of 5}
(ii) R = {(a, b) : a = b}
are equivalence relations. Find the set of all elements related to 2 in each case.

Solution

(ii)

Q5 Find the domain of the following real functions -
(i) f(x) =

x² -4 / x+2

(ii) f(x) =

3x / 28 - x

(iii) f(x) =

x / x²+ 3x + 2

(iv) f(x) =

x³ - x + 3 / x²- 1

Solution

Q6 Find the domain and the range of the following functions :
(i) f(x) =

4-x / √x-4

(ii) f(x) =

x² - 9 / √x-3

(iii) f(x) =

x + 1 / 2x + 1

(iv) f(x)=

4+x / 4-x

Solution

Q7 Show that the function f: I→I defined by f(x) = x² + x is neither one-one nor onto .

Solution

Q8 If the function f: I -> I is defined by
find,
(i) f (3)
(ii) f (4)
(iii) x, if f(x) = -9.
Is this function one-one?

Solution

Q9 Show that the function f: N → N defined by
find,
is both one-one and onto.

Solution

Q10 Show that the function f : R-> R defined by = 2x-1/3 is one-one and onto. Also find the inverse of the function f.

Solution

Q11 If A = R -{-3} and B = R - {2} and a function f: A --> B be given by f(x) =2x-1/x+3,then show that the function f is one-one and onto. Hence, find f^-1

Solution

Q12 Consider the function f :R₊-> [9,∞) defined by f (x) = x² + 9, where R₊ is the set of all non-negative real numbers. Show that f is inversible. Also find the inverse of f

Solution