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Maths Class 11 chapter **“Relations
and Functions”** explains Cartesian product of sets, the
definition
of relation, function as a special kind of relation from one set to another, domain
and
range of functions; sum, difference, product, and quotients of functions, pictorial
representation of a function, ordered pairs; domain, co-domain and range of a
relation,
number of elements in the Cartesian product of two finite sets, the real-valued
function
of the real variable, pictorial diagrams, Cartesian product of the reals with
itself;
domain, co-domain and range of a function; signum and greatest integer functions
with
their graphs; polynomial, constant, modulus, identity, rational, and much more.

##### Question 1:

If (x + 1, y – 2) = (3, 1), find the values of x and y.

##### Answer:

Since the ordered pairs (x + 1, y – 2) and (3, 1)

are equal,

$\therefore $ x + 1 = 3 and y – 2 = 1

$\Rightarrow $ x = 2 and y = 3.

Hence, x = 2 and y = 3.

##### Question 2:

If P = {a, b, c} and Q = {r}, form the sets P × Q and Q × P. Are these products equal

##### Answer:

We have : P = {a, b, c} and Q = {r}.

$\therefore $ P × Q = {(a, r), (b, r), (c, r)}

and Q × P = {(r, a), (r, b), (r, c)}.

Clearly P × Q$\ne $ Q × P. [$\because $(a, r)$\ne $(r,a);etc.]

##### Question 3:

Let A = {1, 2, 3}, B = {3, 4} and
C = {4, 5, 6}. Find :

A × (B $\cap $ C)

##### Answer:

Here B $\cap $ C = {3, 4} $\cap $ {4, 5, 6} = {4}.

$\therefore $ A × (B $\cap $ C) = {1, 2, 3} × {4}

= {(1, 4), (2, 4), (3, 4)}.

##### Question 4:

Let A = {1, 2, 3}, B = {3, 4} and

C = {4, 5, 6}. Find :

(A × B) $\cap $ (A × C)

##### Answer:

Here A × B = {(1, 3), (1, 4), (2, 3), (2, 4),

(3, 3), (3, 4) }

and A × C = {(1, 4), (1, 5), (1, 6), (2, 4),

(2, 5), (2, 6), (3, 4), (3, 5), (3, 6)}.

$\therefore $ (A × B) $\cap $ (A × C) = {(1, 4), (2, 4) × (3, 4)}.

##### Question 5:

Let A = {1, 2, 3}, B = {3, 4} and

C = {4, 5, 6}. Find :

A × (B $\mathrm{\upsilon}$ C)

##### Answer:

Here B $\mathrm{\upsilon}$ C = {3, 4}$\mathrm{\upsilon}${4, 5, 6} = {3, 4, 5, 6}.

$\therefore $ A × (B $\mathrm{\upsilon}$ C) = {(1, 3), (1, 4), (1, 5), (1, 6),

(2, 3), (2, 4), (2, 5), (2, 6) (3, 3), (3, 4), (3, 5), (3, 6)}.

##### Question 6:

Let A = {1, 2, 3}, B = {3, 4} and

C = {4, 5, 6}. Find :

(A × B) $\mathrm{\upsilon}$ (A × C)

##### Answer:

(A × B) $\mathrm{\upsilon}$ (A × C) = {(1, 3), (1, 4), (1, 5), (1, 6),

(2, 3), (2, 4), (2, 5), (2, 6), (3, 3), (3, 4), (3, 5), (3, 6)}.

##### Question 7:

If P = {1, 2}, form the set P × P × P.

##### Answer:

We have : {1, 2}.

$\therefore $ P × P × P = {(1, 1, 1), (1, 1, 2), (1, 2, 1),

(2, 1, 1), (1, 2, 2), (2, 1, 2), (2, 2, 1), (2, 2, 2)}.

##### Question 8:

If R is the set of all real numbers, what do the Cartesian products R × R and R × R × R represent.

##### Answer:

R × R = {(x, y) : x, y $\in $ R}, which represents
the co-ordinates of all points in 2-dimensional space.

(ii) R × R × R = {(x, y, z) ; x, y, z $\in $ R}, which represents
the co-ordinates of all points in 3-dimensional space.

##### Question 9:

If A × B = {(p, q), (p, r), (m, q), (m, r)}, find A and B.

##### Answer:

We have A × B = {(p, q), (p, r),(m, q), (m, r)}.

$\therefore $ A = set of first elements = {p, m}

and B = set of second elements = {q, r}.

##### Question 10:

Let A = {1, 2, 3, 4, 5, 6}. Define a
relation R from A to A by

R = {(x, y) : y = x + 1}.

Depict this relation by arrow diagram.

##### Answer:

By the question, the relation,

R = {(1, 2), (2, 3), (3, 4), (4, 5), (5, 6)}.

The arrow diagram is as shown in the figure.

##### Question 11:

Let A = {1, 2, 3, 4, 5, 6}. Define a
relation R from A to A by

R = {(x, y) : y = x + 1}.

Write down the domain, co-domain and
range of R.

##### Answer:

Domain = {1, 2, 3, 4, 5}

Range = {2, 3, 4, 5, 6}

and Co-domain = {1, 2, 3, 4, 5, 6}.

##### Question 12:

The figure shows a relation between the sets P and Q. Write this relation (i) in set builder form (ii) in roster form What is its domain and range ?

##### Answer:

Here the relation R is “x is sqaure of y”

(i) In set-builder form :

R = {(x, y) : x is the square of y, x $\in $ P, y $\in $ Q}.

In roster form :

R = {(9, 3), (9, – 3), (4, 2), (4, –2), (25, 5), (25, – 5)}.

(ii) Domain of R = {4, 9, 25}.

(iii) Range of R = {– 2, 2, –3 , 3, – 5, 5}.

##### Question 13:

Set A = {1, 2} and B = {3, 4}. Find the number of relations from A to B.

##### Answer:

We have : A × B = {(1, 3), (1, 4), (2, 3), (2, 4)}.

Since n(A × B) = 4,

$\therefore $ the number of subsets of A × B = 2^{4}.

Hence, the number of relations from A to B = 2^{4} = 16.

##### Question 14:

Let N be the set of natural numbers and
the relation R be defined on N such that :

R = {(x, y) : y = 2x, x, y $\in $ N}.

What is the domain, co-domain and range of R ?

Is this relation a function ?

##### Answer:

Domain of R = N, the set of natural numbers.

Co-domain of R = N, the set of natural numbers.

Range of R = set of even natural numbers.

Since each natural number n has one and only one image,

$\therefore $ the relation R is a function.

##### Question 15:

Examine each of the following relations
given below and state in each case, giving reasons whether
it is a function or not ?

R = {(2, 1), (3, 1), (4, 2)}

##### Answer:

Since the elements 2, 3, 4 in the domain of
R have unique images,

$\therefore $ R is a function.

##### Question 16:

Examine each of the following relations
given below and state in each case, giving reasons whether
it is a function or not ?

R = {(2, 2), (2, 4), (3, 3), (4, 4)}

##### Answer:

Since the element 2 in the domain of R has two images
2 and 4,

$\therefore $ R is not a function.

##### Question 17:

Examine each of the following relations
given below and state in each case, giving reasons whether
it is a function or not ?

R = {(1, 2), (2, 3), (3, 4), (4, 5), (5, 6), (6, 7)}

##### Answer:

Since each element in the domain of R has a unique
image,

$\therefore $ R is a function.

##### Question 18:

Let N be the set of natural numbers. Define a real valued function f : N $\to $ N by 2x + 1. Using this definition, complete the table given below :

##### Answer:

The completed table is given by :

[$\because $ f(1) = 2 (1) + 1 = 3, f(2) = 2(2) + 1 = 5; etc.]

##### Question 19:

Define the function f : R$\to $R by f (x) = x^{2}, x $\in $ R. Complete the table given below by using this definition.
What is the domain and range of this function ?

Draw the graph of f.

##### Answer:

The complete table is as below :

Domain of f = {x : x $\in $ R}.

Range of f = {x : x ≥ 0, x $\in $ R}.

The graph of f is as in the adjoining figure.

##### Question 20:

Draw the graph of the function :

f : R $\to $ R defined by f (x) = x^{3}, x $\in $ R.

##### Answer:

We have the following table :

##### Question 21:

Define the real valued function f : R – {0} $\to $ R defined by :

Complete the table given below using this defnition.

What is the domain and range of this function ?

##### Answer:

The complete table is as below.

Domain of f = R – {0}.

Range of f = R – {0}.

The graph of f is as shown in the following figure :

##### Question 22:

Let f(x) = x^{2} and g(x) = 2x + 1 be two
real functions. Find :

##### Answer:

##### Question 23:

##### Answer:

##### Question 24:

Let R be the set of real numbers. Define a real function f : R $\to $ R by f (x) = x + 10. Sketch the graph of this function.

##### Answer:

We have the following table :

The graph of the function is as shown below :

##### Question 25:

Let R be a relation from Q to Q defined by :

R = {(a, b) : a, b $\in $ Q and a – b $\in $ Z}.

Show that : (a, a) $\in $ R for all Q $\in $ Q

##### Answer:

Since a – a = 0 $\in $ Z, therefore, (a, a) $\in $ R.

##### Question 26:

Let R be a relation from Q to Q defined by :

R = {(a, b) : a, b $\in $ Q and a – b $\in $ Z}.

Show that : (a, b) $\in $ R implies that (b, a) $\in $ R

##### Answer:

(a, b) $\in $ R $\Rightarrow $ a – b $\in $ Z $\Rightarrow $ b – a $\in $ Z $\Rightarrow $ (b, a) $\in $ R.

##### Question 27:

Let R be a relation from Q to Q defined by :

R = {(a, b) : a, b $\in $ Q and a – b $\in $ Z}.

Show that : (a, b) $\in $ R and (b, c) $\in $ R implies that (a, c) $\in $ R

##### Answer:

(a, b) $\in $ R and (b, c) $\in $ R

$\Rightarrow $ a – b $\in $ Z and b – c $\in $ Z

$\Rightarrow $ (a – b) + (b – c) = a – c $\in $ Z.

Hence, (a, c) $\in $ R.

##### Question 28:

Let f = {(1, 1), (2, 3), (0, – 1), (– 1, – 3)} be a linear function from Z to Z. Find f(x).

##### Answer:

Since ‘f’ is linear function,

$\therefore $ let f(x) = mx + c ...(1)

Now (1, 1) $\in $ R $\Rightarrow $ 1 = m + c $\Rightarrow $ m + c = 1 ...(2)

And (0, – 1) $\in $ R $\Rightarrow $ – 1 = 0 + c $\Rightarrow $ c = – 1.

Putting in (2), m – 1 $\Rightarrow $ m = 2.

Putting in (1), f(x) = 2x – 1.

##### Question 29:

Find the domain of the function :

##### Answer:

Now x^{2} – 5x + 4 = (x – 4) (x – 1).

$\therefore $ x^{2} – 5x + 4 = 0 $\Rightarrow $ x = 4, 1.

Thus, that function ‘f’ is defined for all real value except

at x = 4, 1.

Hence, domain of f = R – {1, 4}.

##### Question 30:

The function ‘f’ is defined by :

Draw the graph of f (x).

##### Answer:

We have : f (x) = 1– x, x < 0.

$\therefore $ f (– 4) = 1 – (– 4) = 1 + 4 = 5

f (–3) = 1 – (–3) = 1 + 3 = 4

f (–2) = 1 – (–2) = 1 + 2 = 3

f (–1) = 1 – (–1) = 1 + 1 = 2.

Also, f (0) = 1.

And f (x) = x + 1, x > 0.

$\therefore $ f (1) = 1 + 1 = 2, f (2) = 2 + 1 = 3,

f (3) = 3 + 1 = 4, f (4) = 4 + 1 = 5 ; .....

Hence, the graph of f is as shown :

##### Question 31:

##### Answer:

##### Question 32:

If the set A has 3 elements and the set B =

{3, 4, 5}, then find the number of elements in

(A × B).

##### Answer:

A has 3 elements and B has also 3 elements.

$\therefore $ Number of elements in (A × B) = 3 × 3 = 9.

##### Question 33:

If G = {7, 8}, H = {5, 4, 2}, find G × H and H × G.

##### Answer:

(i) G × H = {7, 8} × {5, 4, 2}

= {(7, 5), (7, 4), (7, 2), (8, 5), (8, 4), (8, 2)}.

(ii) H × G = {5, 4, 2} × {7, 8}

= {(5, 7), (5, 8), (4, 7), (4, 8), (2, 7), (2, 8)}.

##### Question 34:

State whether each of the following statements are
true or false. If the statement is false, rewrite the given
statement correctly.

If P = {m, n}

Q = {n, m},

then P × Q = {(m, n), (n, m)}.

##### Answer:

False.

P × Q = {(m, n), (m, m), (n, m) (n, n)}.

##### Question 35:

State whether each of the following statements are
true or false. If the statement is false, rewrite the given
statement correctly.

If A and B are non-empty sets, then A × B is nonempty
set of ordered pairs (x, y) such that x $\in $ Β and
y $\in $ A.

##### Answer:

False.

A × B is a non-empty set of ordered pairs.

(x, y) such that x $\in $ A and y $\in $ B.

##### Question 36:

State whether each of the following statements are
true or false. If the statement is false, rewrite the given
statement correctly.

If A = {1, 2}, B = {3, 4}, then :

A × (B $\cap $ $\mathrm{\varphi}$) = $\mathrm{\varphi}$.

##### Answer:

True.

[$\because $ B $\cap $ $\mathrm{\varphi}$ = $\mathrm{\varphi}$, $\therefore $ A × (B $\cap $ $\mathrm{\varphi}$) = A × $\mathrm{\varphi}$ = $\mathrm{\varphi}$]

##### Question 37:

If A = {– 1, 1}, find A × A × A.

##### Answer:

A × A × A ={– 1, 1} × {– 1, 1} × {– 1, 1}

= {(– 1, – 1, – 1), (– 1, – 1, 1),

(– 1, 1, – 1), (– 1, 1, 1), (1, – 1, – 1),

(1, – 1, 1), (1, 1, – 1), (1, 1, 1)}.

##### Question 38:

If A × B = [(a, x), (a, y), (b, x), (b, y)}, find A and B.

##### Answer:

We have :

A = {(a, x), (a, y), (b, x), (b, y)}.

$\therefore $ A = {a, b} and B = {x, y}.

##### Question 39:

Let A = {1, 2}, B = {1, 2, 3, 4}, C = {5, 6} and

D = {5, 6, 7, 8].

Verify that :

A × (B $\cap $ C) = (A × B) $\cap $ (A × C)

##### Answer:

B $\cap $ C = {1, 2, 3, 4} $\cap $ {5, 6} = $\mathrm{\varphi}$.

$\therefore $ LHS = A × (B $\cap $ C) = A × $\mathrm{\varphi}$

= {1, 2} × $\mathrm{\varphi}$ = $\mathrm{\varphi}$.

Now A × B = {1, 2} × {1, 2, 3, 4}

= {(1, 1), (1, 2), (1, 3), (1, 4), (2, 1), (2, 2), (2, 3), (2, 4)}

and A × C = {1, 2} × {5, 6}

= {(1, 5), (1, 6), (2, 5), (2, 6)}.

$\therefore $ RHS = (A × B) $\cap $ (A × C) = $\mathrm{\varphi}$.

Hence, A × (B $\cap $ C) = (A × B) $\cap $ (A × C).

##### Question 40:

Let A = {1, 2}, B = {1, 2, 3, 4}, C = {5, 6} and

D = {5, 6, 7, 8].

Verify that :

A × C is a subset of B × D.

##### Answer:

A × C = {(1, 5), (1, 6), (2, 5), (2, 6)}

[As in part (i)]

B × D = {1, 2, 3, 4} × {5, 6, 7, 8}

= {(1, 5), (1, 6), (1, 7), (1, 8), (2, 5), (2, 6),

(2, 7), (2, 8), (3, 5), (3, 6), (3, 7), (3, 8),

(4, 5), (4, 6), (4, 7), (4, 8)}.

Clearly each element of A × C is in B × D.

Hence, A × C is a subset of B × D.

##### Question 41:

Let A = {1, 2} and B = {3, 4}. Write A × B. How many subsets will A × B have ? List them.

##### Answer:

A × B = {1, 2} × {3, 4}

= {(1, 3), (1, 4), (2, 3), (2, 4)}.

No. of subsets of A × B = 2^{4} = 16.

These subsets are :

$\mathrm{\varphi}$, {(1, 3)}, {(1, 4)}, {(2, 3)},

{(2, 4)},{(1, 3), (1,4)}, {(1, 3), (2, 3)}, {(1, 3), (2, 4},

{(1, 4), (2, 3)}, {((1, 4), (2, 4)}, {(2, 3), (2, 4)},

{(1, 3), (1, 4), (2, 3)}, {(1, 4), (2, 3), (2, 4)},

{((2, 3), (2, 4), (1, 3))}, {(1, 3), (1, 4), (2, 4)},

{(1, 3), (1, 4), (2, 3), (2, 4)}.

##### Question 42:

Let A and B be two sets such that n (A) = 3, 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.

##### Answer:

Since x, y, z $\in $ A,

$\therefore $ A = {x, y, z}.

Since 1, 2 $\in $ B,

$\therefore $ B = {1, 2}.

##### Question 43:

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.

##### Answer:

Here (– 1, 0) $\in $ A × A

and (0, 1) $\in $ A × A.

$\therefore $ – 1, 0 $\in $ A

and 0, 1 $\in $ A.

Thus A = {– 1, 0, 1}.

$\therefore $ A × A = {– 1, 0, 1} × {– 1, 0, 1}

= {(– 1, – 1), (– 1, 0), (– 1, 1), (0, – 1),

(0, 0), (0, 1), (1, – 1), (1, 0), (1, 1)}.

$\therefore $ Remaining elements of A × A are :

(– 1, – 1), (– 1, 1), (0, – 1), (0, 0), (1, – 1), (1, 0), (1, 1).

##### Question 44:

Let A = {1, 2, 3, ........., 14}. Define a relation R from A to A by R = {(x, y) ; 3x – y = 0, where x, y $\in $ A}. Write down its domain, co-domain and range.

##### Answer:

Here R = {(x, y) : 3x – y = 0, where x, y $\in $ A}

= {(1, 3), (2, 6), (3, 9), (4, 12)}.

(i) Domain = {1, 2, 3, 4}

(ii) Co-domain = {1, 2, 3, ......., 14}

(iii) Range = {3, 6, 9, 12}.

##### Question 45:

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 $\in $ N}.

Depict this relationship using (i) roster form (ii) an
arrow diagram. Write down the domain and range.

##### Answer:

##### Question 46:

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 $\in $ A, y $\in $ B}. Write R in roster form.

##### Answer:

Here R = {(1, 4), (1, 6), (2, 9), (3, 4),

(3, 6), (5, 4), (5, 6)}

[$\because $ 4 – 1 = 3; odd etc.]

##### Question 47:

Figure given below shows a relationship between the sets P and Q. Write the relation (i) in set builder form (ii) roster form. What is its domain and range ?

##### Answer:

(i) R = {(x, y) : x – y = 2, 4 < x < 8, x, y $\in $ N}

(ii) R = {(5, 3), (6, 4), (7, 5)}.

Domain = {5, 6, 7}. Range = { 3, 4, 5}.

##### Question 48:

Write the relation ?

R = {(x, x^{3}) : x is a prime number less than 10}
in roster form.

##### Answer:

Prime numbers less than 10 are 2, 3, 5, 7.

$\therefore $ R = {(2, 8), (3, 27), (5, 123), (7, 343)}.

##### Question 49:

Let A = {x, y, z} and B = {1, 2}. Find the number of relations from A into B.

##### Answer:

We have : A = {x, y, z}

and B = {1, 2}.

$\therefore $ A × B = {(x, 1), (x, 2), (y, 1), (y, 2), (z, 1), (z, 2)}.

$\therefore $ n (A × B) = 6

$\Rightarrow $ the number of subsets of n (A × B) = 2^{6}.

Hence, the number of relation from A into B = 2^{6}.

##### Question 50:

Let R be the relation on Z defined by :

R = {(a, b) : a, b $\in $ Z, a – b is an integer}

Find the domain and range of R.

##### Answer:

We have : R = {(a, b) : a, b $\in $ Z,

a – b is an integer}

$\therefore $ Domain of R = Z.

Range of R = Z.

##### Question 51:

Which of the following relations are functions ? Give
reasons. If it is a function, determine its domain and
range :

{(2, 1), (5, 1), (8, 1) (11,1), (14, 1), (17, 1)}

##### Answer:

We have :

f = {(2, 1), (5, 1), (8, 1), (11, 1), (14, 1), (17, 1)}.

Here no two ordered pairs have the same first component.

$\therefore $ this relation is a function.

Domain of f = {2, 5, 8, 11, 14, 17}.

Range of f = {1}.

##### Question 52:

Which of the following relations are functions ? Give
reasons. If it is a function, determine its domain and
range :

{(2, 1), (4, 2), (6, 3), (8, 4), (10, 5), (12, 6), (14, 7)}

##### Answer:

We have : f = {(2, 1), (4, 2), (6, 3), (8, 4),
(10, 5), (12, 6), (14, 7)}.

Here no two ordered pairs have the same first component.
$\therefore $ this relation is a function.

Domain of f = {2, 4, 6, 8, 10, 12, 14}.

Range of f = {1, 2, 3, 4, 5, 6, 7}.

##### Question 53:

Which of the following relations are functions ? Give
reasons. If it is a function, determine its domain and
range :

{(1, 3), (1, 5), (2, 5)}

##### Answer:

We have : f = {(1, 3), (1, 5), (2,5)}.
Here 1 appears more than once as a first component in the
ordered pairs of f.

Hence, this relation is not a function.

##### Question 54:

Find the domain and range of the following real functions :

f (x) = – | x |

##### Answer:

We have : f (x) = – | x |.

Clearly f (x) $\le $ 0 ∀ x $\in $ R.

Domain of f = R.

Range of f = (– $\infty $, 0].

##### Question 55:

Find the domain and range of the following real functions :

##### Answer:

##### Question 56:

A function f is defined by f (x) = 2x – 5. Write down
the values of :

f (0)

##### Answer:

We have : f (x) = 2x – 5.

f (0) = 2 (0) – 5 = 0 – 5 = – 5.

##### Question 57:

A function f is defined by f (x) = 2x – 5. Write down
the values of :

f (7)

##### Answer:

We have : f (x) = 2x – 5.

f (7) = 2 (7) – 5 = 14 – 5 = 9.

##### Question 58:

A function f is defined by f (x) = 2x – 5. Write down
the values of :

f (– 3)

##### Answer:

We have : f (x) = 2x – 5.

f (– 3) = 2 (– 3) – 5 = – 6 – 5 = – 11.

##### Question 59:

The function ‘f’, which maps temperature in degree celsius into temperature degree Fahrenheit is defined by

Find :

t (0)

##### Answer:

Putting C = 0, t (0) = 0 + 32 = 32.

##### Question 60:

The function ‘f’, which maps temperature in degree celsius into temperature degree Fahrenheit is defined by

Find :

t (28)

##### Answer:

##### Question 61:

The function ‘f’, which maps temperature in degree celsius into temperature degree Fahrenheit is defined by

Find :

t (– 10)

##### Answer:

##### Question 62:

Find :

The value of C, when t (C) = 212

##### Answer:

##### Question 63:

Find the range of each of the following function :

f (x) = 2 – 3x, x $\in $ R, x > 0

##### Answer:

##### Question 64:

Find the range of each of the following function :

f (x) = x^{2} + 2, x is a real number

##### Answer:

##### Question 65:

Find the range of each of the following function :

f (x) = x, x is a real number

##### Answer:

We have f (x) = x $\Rightarrow $ y = x.

But x is a real number

$\Rightarrow $y is a real number.

Hence, Range of f = {y : y $\in $ R}

= R.