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Relations and Functions
NCERT Solutions
NCERT Solutions
Relations and Functions
36 Solutions
Exercise:
All Exercises
EXERCISE 2.1
EXERCISE 2.2
EXERCISE 2.3
Miscellaneous Exercise on Chapter 2
Q1
EXERCISE 2.1
If
(
x
3
+
1
,
y
−
2
3
)
=
(
5
3
,
1
3
)
(\frac{x}{3}+1, y-\frac{2}{3})=(\frac{5}{3}, \frac{1}{3})
(
3
x
+
1
,
y
−
3
2
)
=
(
3
5
,
3
1
)
, find the values of
x
x
x
and
y
y
y
.
Q2
EXERCISE 2.1
If the set A has 3 elements and the set B = {3, 4, 5}, then find the number of elements in (A × B).
Q3
EXERCISE 2.1
If G = {7, 8} and H = {5, 4, 2}, find G × H and H × G.
Q4
EXERCISE 2.1
State whether each of the following statements are true or false. If the statement is false, rewrite the given statement correctly.
(i)
If P = {m, n} and Q = {n, m}, then P × Q = {(m, n), (n, m)}.
(ii)
If A and B are non-empty sets, then A × B is a non-empty set of ordered pairs (x, y) such that x ∈ A and y ∈ B.
(iii)
If A = {1, 2}, B = {3, 4}, then A × (B ∩ φ) = φ.
Q5
EXERCISE 2.1
If A = {-1, 1}, find A × A × A.
Q6
EXERCISE 2.1
If A × B = {(a, x), (a, y), (b, x), (b, y)}. Find A and B.
Q7
EXERCISE 2.1
Let A = {1, 2}, B = {1, 2, 3, 4}, C = {5, 6} and D = {5, 6, 7, 8}. Verify that
(i)
A × (B ∩ C) = (A × B) ∩ (A × C).
(ii)
A × C is a subset of B × D.
Q8
EXERCISE 2.1
Let A = {1, 2} and B = {3, 4}. Write A × B. How many subsets will A × B have? List them.
Q9
EXERCISE 2.1
Let A and B be two sets such that n(A) = 3 and n(B) = 2. If (x, 1), (y, 2), (z, 1) are in A × B, find A and B, where x, y and z are distinct elements.
Q10
EXERCISE 2.1
The Cartesian product A × A has 9 elements among which are found (-1, 0) and (0, 1). Find the set A and the remaining elements of A × A.
Q1
EXERCISE 2.2
Let A = {1, 2, 3, ..., 14}. Define a relation R from A to A by R = {(x, y): 3x - y = 0, where x, y ∈ A}. Write down its domain, codomain and range.
Q2
EXERCISE 2.2
Define a relation R on the set N of natural numbers by R = {(x, y): y = x + 5, x is a natural number less than 4; x, y ∈ N}. Depict this relationship using roster form. Write down the domain and the range.
Q3
EXERCISE 2.2
A = {1, 2, 3, 5} and B = {4, 6, 9}. Define a relation R from A to B by R = {(x, y): the difference between x and y is odd; x ∈ A, y ∈ B}. Write R in roster form.
Q4
EXERCISE 2.2
The Fig2. 7 shows a relationship between the sets P and Q. Write this relation (i) in set-builder form (ii) roster form. What is its domain and range?
Q5
EXERCISE 2.2
Let A = {1, 2, 3, 4, 6}. Let R be the relation on A defined by {(a, b): a, b ∈ A, b is exactly divisible by a}.
(i)
Write R in roster form
(ii)
Find the domain of R
(iii)
Find the range of R.
Q6
EXERCISE 2.2
Determine the domain and range of the relation R defined by R = {(x, x + 5): x ∈ {0, 1, 2, 3, 4, 5}}.
Q7
EXERCISE 2.2
Write the relation R = {(x, x³): x is a prime number less than 10} in roster form.
Q8
EXERCISE 2.2
Let A = {x, y, z} and B = {1, 2}. Find the number of relations from A to B.
Q9
EXERCISE 2.2
Let R be the relation on Z defined by R = {(a, b): a, b ∈ Z, a - b is an integer}. Find the domain and range of R.
Q1
EXERCISE 2.3
Which of the following relations are functions? Give reasons. If it is a function, determine its domain and range.
(i)
{(2,1), (5,1), (8,1), (11,1), (14,1), (17,1)}
(ii)
{(2,1), (4,2), (6,3), (8,4), (10,5), (12,6), (14,7)}
(iii)
{(1,3), (1,5), (2,5)}
Q2
EXERCISE 2.3
Find the domain and range of the following real functions:
(i)
f(x) = -|x|
(ii)
f(x) = √(9-x²)
Q3
EXERCISE 2.3
A function f is defined by f(x) = 2x - 5. Write down the values of
(i)
f(0),
(ii)
f(7),
(iii)
f(-3).
Q4
EXERCISE 2.3
The function 't' which maps temperature in degree Celsius into temperature in degree Fahrenheit is defined by t(C) = (9C/5) + 32.
Find (i) t(0) (ii) t(28) (iii) t(-10) (iv) The value of C, when t(C) = 212.
Q5
EXERCISE 2.3
Find the range of each of the following functions.
(i)
f(x) = 2 - 3x, x ∈ R, x > 0.
(ii)
f(x) = x² + 2, x is a real number.
(iii)
f(x) = x, x is a real number.
Q1
Miscellaneous Exercise on Chapter 2
The relation f is defined by
f
(
x
)
=
{
x
2
,
0
≤
x
≤
3
3
x
,
3
≤
x
≤
10
f(x)=\begin{cases}x^{2}, 0 \leq x \leq 3 \ 3 x, 3 \leq x \leq 10\end{cases}
f
(
x
)
=
{
x
2
,
0
≤
x
≤
3
3
x
,
3
≤
x
≤
10
The relation g is defined by
g
(
x
)
=
{
x
2
,
0
≤
x
≤
2
3
x
,
2
≤
x
≤
10
g(x)=\begin{cases}x^{2}, 0 \leq x \leq 2 \ 3 x, 2 \leq x \leq 10\end{cases}
g
(
x
)
=
{
x
2
,
0
≤
x
≤
2
3
x
,
2
≤
x
≤
10
Show that f is a function and g is not a function.
Q2
Miscellaneous Exercise on Chapter 2
If
f
(
x
)
=
x
2
f(x)=x^{2}
f
(
x
)
=
x
2
, find
f
(
1.1
)
−
f
(
1
)
(
1.1
−
1
)
\frac{f(1.1)-f(1)}{(1.1-1)}
(
1.1
−
1
)
f
(
1.1
)
−
f
(
1
)
.
Q3
Miscellaneous Exercise on Chapter 2
Find the domain of the function
f
(
x
)
=
x
2
+
2
x
+
1
x
2
−
8
x
+
12
f(x)=\frac{x^{2}+2 x+1}{x^{2}-8 x+12}
f
(
x
)
=
x
2
−
8
x
+
12
x
2
+
2
x
+
1
.
Q4
Miscellaneous Exercise on Chapter 2
Find the domain and the range of the real function f defined by
f
(
x
)
=
(
x
−
1
)
f(x)=\sqrt{(x-1)}
f
(
x
)
=
(
x
−
1
)
.
Q5
Miscellaneous Exercise on Chapter 2
Find the domain and the range of the real function f defined by
f
(
x
)
=
∣
x
−
1
∣
f(x)=|x-1|
f
(
x
)
=
∣
x
−
1∣
.
Q6
Miscellaneous Exercise on Chapter 2
Let
f
=
{
(
x
,
x
2
1
+
x
2
)
:
x
∈
R
}
f=\left\{\left(x, \frac{x^{2}}{1+x^{2}}\right): x \in \mathbf{R}\right\}
f
=
{
(
x
,
1
+
x
2
x
2
)
:
x
∈
R
}
be a function from R into R. Determine the range of f.
Q7
Miscellaneous Exercise on Chapter 2
Let
f
,
g
:
R
→
R
f, g: R \rightarrow R
f
,
g
:
R
→
R
be defined, respectively by
f
(
x
)
=
x
+
1
,
g
(
x
)
=
2
x
−
3
f(x)=x+1, g(x)=2x-3
f
(
x
)
=
x
+
1
,
g
(
x
)
=
2
x
−
3
. Find
f
+
g
,
f
−
g
f+g, f-g
f
+
g
,
f
−
g
and
f
g
\frac{f}{g}
g
f
.
Q8
Miscellaneous Exercise on Chapter 2
Let
f
=
{
(
1
,
1
)
,
(
2
,
3
)
,
(
0
,
−
1
)
,
(
−
1
,
−
3
)
}
f=\{(1,1),(2,3),(0,-1),(-1,-3)\}
f
=
{(
1
,
1
)
,
(
2
,
3
)
,
(
0
,
−
1
)
,
(
−
1
,
−
3
)}
be a function from Z to Z defined by
f
(
x
)
=
a
x
+
b
f(x)=ax+b
f
(
x
)
=
a
x
+
b
, for some integers a, b. Determine a, b.
Q9
Miscellaneous Exercise on Chapter 2
Let R be a relation from N to N defined by R = {(a, b): a, b ∈ N and a = b²}. Are the following true?
(i)
(a, a) ∈ R, for all a ∈ N
(ii)
(a, b) ∈ R, implies (b, a) ∈ R
(iii)
(a, b) ∈ R, (b, c) ∈ R implies (a, c) ∈ R. Justify your answer in each case.
Q10
Miscellaneous Exercise on Chapter 2
Let A = {1, 2, 3, 4}, B = {1, 5, 9, 11, 15, 16} and f = {(1, 5), (2, 9), (3, 1), (4, 5), (2, 11)}. Are the following true?
(i)
f is a relation from A to B
(ii)
f is a function from A to B. Justify your answer in each case.
Q11
Miscellaneous Exercise on Chapter 2
Let f be the subset of Z × Z defined by f = {(ab, a+b): a, b ∈ Z}. Is f a function from Z to Z? Justify your answer.
Q12
Miscellaneous Exercise on Chapter 2
Let A = {9, 10, 11, 12, 13} and let f: A → N be defined by f(n) = the highest prime factor of n. Find the range of f.
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