NCERT Textbook - Relations and Functions

# NCERT Textbook - Relations and Functions | NCERT Textbooks (Class 6 to Class 12) - CTET & State TET PDF Download

``` Page 1

vMathematics is the indispensable instrument of
all physical research. – BERTHELOT v
2.1  Introduction
Much of mathematics is about finding a pattern – a
recognisable link between quantities that change. In our
daily life, we come across many patterns that characterise
relations such as brother and sister, father and son, teacher
and student. In mathematics also, we come across many
relations such as number m is less than number n, line l is
parallel to line m, set A is a subset of set B. In all these, we
notice that a relation involves pairs of objects in certain
order. In this Chapter, we will learn how to link pairs of
objects from two sets and then introduce relations between
the two objects in the pair. Finally, we will learn about
special relations which will qualify to be functions. The
concept of function is very important in mathematics since it captures the idea of a
mathematically precise correspondence between one quantity with the other.
2.2  Cartesian Products of Sets
Suppose A is a set of 2 colours and B is a set of  3 objects, i.e.,
A = {red, blue}and B = {b, c, s},
where b, c and s represent a particular bag, coat and shirt, respectively.
How many pairs of coloured objects can be made from these two sets?
Proceeding in a very orderly manner, we can see that there will be 6
distinct pairs as given below:
(red, b), (red, c), (red, s), (blue, b), (blue, c), (blue, s).
Thus, we get 6 distinct objects (Fig 2.1).
Let us recall from our earlier classes that an ordered pair of elements
taken from any two sets P and Q is a pair of elements written in small
Fig 2.1
Chapter 2
RELATIONS AND FUNCTIONS
G . W.  Leibnitz
(1646–1716)
Rationalised 2023-24
Page 2

vMathematics is the indispensable instrument of
all physical research. – BERTHELOT v
2.1  Introduction
Much of mathematics is about finding a pattern – a
recognisable link between quantities that change. In our
daily life, we come across many patterns that characterise
relations such as brother and sister, father and son, teacher
and student. In mathematics also, we come across many
relations such as number m is less than number n, line l is
parallel to line m, set A is a subset of set B. In all these, we
notice that a relation involves pairs of objects in certain
order. In this Chapter, we will learn how to link pairs of
objects from two sets and then introduce relations between
the two objects in the pair. Finally, we will learn about
special relations which will qualify to be functions. The
concept of function is very important in mathematics since it captures the idea of a
mathematically precise correspondence between one quantity with the other.
2.2  Cartesian Products of Sets
Suppose A is a set of 2 colours and B is a set of  3 objects, i.e.,
A = {red, blue}and B = {b, c, s},
where b, c and s represent a particular bag, coat and shirt, respectively.
How many pairs of coloured objects can be made from these two sets?
Proceeding in a very orderly manner, we can see that there will be 6
distinct pairs as given below:
(red, b), (red, c), (red, s), (blue, b), (blue, c), (blue, s).
Thus, we get 6 distinct objects (Fig 2.1).
Let us recall from our earlier classes that an ordered pair of elements
taken from any two sets P and Q is a pair of elements written in small
Fig 2.1
Chapter 2
RELATIONS AND FUNCTIONS
G . W.  Leibnitz
(1646–1716)
Rationalised 2023-24
RELATIONS AND FUNCTIONS          25
brackets and grouped together in a particular order, i.e., (p,q), p ? P and  q ? Q . This
Definition 1 Given two non-empty sets P and Q. The cartesian product P × Q is the
set of all ordered pairs of elements from P and Q, i.e.,
P × Q = { (p,q) : p  ? P, q  ? Q }
If either P or Q is the null set, then P × Q will also be empty set, i.e., P × Q = f
From the illustration given above we note that
A × B = {(red,b), (red,c), (red,s), (blue,b), (blue,c), (blue,s)}.
Again, consider the two sets:
A = {DL, MP, KA}, where DL, MP, KA represent Delhi,
03}representing codes for the licence plates of vehicles issued
by DL, MP and KA .
were making codes for the licence plates of vehicles, with the
restriction that the code begins with an element from set A,
which are the pairs available from these sets and how many such
pairs will there be (Fig 2.2)?
The available pairs are:(DL,01), (DL,02), (DL,03), (MP,01), (MP,02), (MP,03),
(KA,01), (KA,02), (KA,03) and the product of set A and set B is given by
A × B = {(DL,01), (DL,02), (DL,03), (MP,01), (MP,02), (MP,03), (KA,01), (KA,02),
(KA,03)}.
It can easily be seen that there will be 9 such pairs in the Cartesian product, since
there are 3 elements in each of the sets A and B. This gives us 9 possible codes. Also
note that the order in which these elements are paired is crucial. For example, the code
(DL, 01) will not be the same as the code (01, DL).
As a final illustration, consider the two sets A= {a
1
, a
2
} and
B = {b
1
, b
2
, b
3
, b
4
} (Fig 2.3).
A × B = {( a
1
, b
1
), (a
1
, b
2
), (a
1
, b
3
), (a
1
, b
4
), (a
2
, b
1
), (a
2
, b
2
),
(a
2
, b
3
), (a
2
, b
4
)}.
The 8 ordered pairs thus formed can represent the position of points in
the plane if A and B are subsets of the set of real numbers and it is
obvious that the point in the position (a
1
, b
2
) will be distinct from the point
in the position (b
2
, a
1
).
Remarks
(i) Two ordered pairs are equal, if and only if  the corresponding first elements
are equal and the second  elements are also equal.
DL MP KA
03
02
01
Fig 2.2
Fig 2.3
Rationalised 2023-24
Page 3

vMathematics is the indispensable instrument of
all physical research. – BERTHELOT v
2.1  Introduction
Much of mathematics is about finding a pattern – a
recognisable link between quantities that change. In our
daily life, we come across many patterns that characterise
relations such as brother and sister, father and son, teacher
and student. In mathematics also, we come across many
relations such as number m is less than number n, line l is
parallel to line m, set A is a subset of set B. In all these, we
notice that a relation involves pairs of objects in certain
order. In this Chapter, we will learn how to link pairs of
objects from two sets and then introduce relations between
the two objects in the pair. Finally, we will learn about
special relations which will qualify to be functions. The
concept of function is very important in mathematics since it captures the idea of a
mathematically precise correspondence between one quantity with the other.
2.2  Cartesian Products of Sets
Suppose A is a set of 2 colours and B is a set of  3 objects, i.e.,
A = {red, blue}and B = {b, c, s},
where b, c and s represent a particular bag, coat and shirt, respectively.
How many pairs of coloured objects can be made from these two sets?
Proceeding in a very orderly manner, we can see that there will be 6
distinct pairs as given below:
(red, b), (red, c), (red, s), (blue, b), (blue, c), (blue, s).
Thus, we get 6 distinct objects (Fig 2.1).
Let us recall from our earlier classes that an ordered pair of elements
taken from any two sets P and Q is a pair of elements written in small
Fig 2.1
Chapter 2
RELATIONS AND FUNCTIONS
G . W.  Leibnitz
(1646–1716)
Rationalised 2023-24
RELATIONS AND FUNCTIONS          25
brackets and grouped together in a particular order, i.e., (p,q), p ? P and  q ? Q . This
Definition 1 Given two non-empty sets P and Q. The cartesian product P × Q is the
set of all ordered pairs of elements from P and Q, i.e.,
P × Q = { (p,q) : p  ? P, q  ? Q }
If either P or Q is the null set, then P × Q will also be empty set, i.e., P × Q = f
From the illustration given above we note that
A × B = {(red,b), (red,c), (red,s), (blue,b), (blue,c), (blue,s)}.
Again, consider the two sets:
A = {DL, MP, KA}, where DL, MP, KA represent Delhi,
03}representing codes for the licence plates of vehicles issued
by DL, MP and KA .
were making codes for the licence plates of vehicles, with the
restriction that the code begins with an element from set A,
which are the pairs available from these sets and how many such
pairs will there be (Fig 2.2)?
The available pairs are:(DL,01), (DL,02), (DL,03), (MP,01), (MP,02), (MP,03),
(KA,01), (KA,02), (KA,03) and the product of set A and set B is given by
A × B = {(DL,01), (DL,02), (DL,03), (MP,01), (MP,02), (MP,03), (KA,01), (KA,02),
(KA,03)}.
It can easily be seen that there will be 9 such pairs in the Cartesian product, since
there are 3 elements in each of the sets A and B. This gives us 9 possible codes. Also
note that the order in which these elements are paired is crucial. For example, the code
(DL, 01) will not be the same as the code (01, DL).
As a final illustration, consider the two sets A= {a
1
, a
2
} and
B = {b
1
, b
2
, b
3
, b
4
} (Fig 2.3).
A × B = {( a
1
, b
1
), (a
1
, b
2
), (a
1
, b
3
), (a
1
, b
4
), (a
2
, b
1
), (a
2
, b
2
),
(a
2
, b
3
), (a
2
, b
4
)}.
The 8 ordered pairs thus formed can represent the position of points in
the plane if A and B are subsets of the set of real numbers and it is
obvious that the point in the position (a
1
, b
2
) will be distinct from the point
in the position (b
2
, a
1
).
Remarks
(i) Two ordered pairs are equal, if and only if  the corresponding first elements
are equal and the second  elements are also equal.
DL MP KA
03
02
01
Fig 2.2
Fig 2.3
Rationalised 2023-24
26 MATHEMATICS
(ii) If there are p elements in A and q elements in B, then there will be pq
elements in A × B, i.e.,  if n(A) = p and n(B) = q,  then n(A × B) = pq.
(iii) If A and B are non-empty sets and either A or B is an infinite set, then so is
A × B.
(iv) A × A × A = {(a, b, c) : a, b, c ? A}. Here (a, b, c) is called an ordered
triplet.
Example 1 If  (x + 1, y – 2) = (3,1), find the values of x and y.
Solution Since the ordered pairs are equal, the corresponding elements are equal.
Therefore x + 1 = 3  and y – 2 = 1.
Solving we get x = 2 and y = 3.
Example 2 If P = {a, b, c} and Q = {r}, form the sets P × Q and Q × P.
Are these two products equal?
Solution By the definition of the cartesian product,
P × Q =  {(a, r), (b, r), (c, r)} and Q × P =  {(r, a), (r, b), (r, c)}
Since, by the definition of equality of ordered pairs, the pair (a, r) is not equal to the pair
(r, a), we conclude that P × Q ? Q × P.
However, the number of elements in each set will be the same.
Example 3 Let A = {1,2,3}, B = {3,4} and C = {4,5,6}. Find
(i) A × (B n C) (ii) (A × B) n (A × C)
(iii) A × (B ? C) (iv) (A × B) ? (A × C)
Solution (i) By the definition of the intersection of two sets, (B n C) = {4}.
Therefore, A × (B n C) = {(1,4), (2,4), (3,4)}.
(ii) Now (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) n (A × C)  = {(1, 4), (2, 4), (3, 4)}.
(iii) Since, (B ? C) = {3, 4, 5, 6}, we have
A × (B ? 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)}.
(iv) Using the sets A × B and A × C from part (ii) above, we obtain
(A × B) ? (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)}.
Rationalised 2023-24
Page 4

vMathematics is the indispensable instrument of
all physical research. – BERTHELOT v
2.1  Introduction
Much of mathematics is about finding a pattern – a
recognisable link between quantities that change. In our
daily life, we come across many patterns that characterise
relations such as brother and sister, father and son, teacher
and student. In mathematics also, we come across many
relations such as number m is less than number n, line l is
parallel to line m, set A is a subset of set B. In all these, we
notice that a relation involves pairs of objects in certain
order. In this Chapter, we will learn how to link pairs of
objects from two sets and then introduce relations between
the two objects in the pair. Finally, we will learn about
special relations which will qualify to be functions. The
concept of function is very important in mathematics since it captures the idea of a
mathematically precise correspondence between one quantity with the other.
2.2  Cartesian Products of Sets
Suppose A is a set of 2 colours and B is a set of  3 objects, i.e.,
A = {red, blue}and B = {b, c, s},
where b, c and s represent a particular bag, coat and shirt, respectively.
How many pairs of coloured objects can be made from these two sets?
Proceeding in a very orderly manner, we can see that there will be 6
distinct pairs as given below:
(red, b), (red, c), (red, s), (blue, b), (blue, c), (blue, s).
Thus, we get 6 distinct objects (Fig 2.1).
Let us recall from our earlier classes that an ordered pair of elements
taken from any two sets P and Q is a pair of elements written in small
Fig 2.1
Chapter 2
RELATIONS AND FUNCTIONS
G . W.  Leibnitz
(1646–1716)
Rationalised 2023-24
RELATIONS AND FUNCTIONS          25
brackets and grouped together in a particular order, i.e., (p,q), p ? P and  q ? Q . This
Definition 1 Given two non-empty sets P and Q. The cartesian product P × Q is the
set of all ordered pairs of elements from P and Q, i.e.,
P × Q = { (p,q) : p  ? P, q  ? Q }
If either P or Q is the null set, then P × Q will also be empty set, i.e., P × Q = f
From the illustration given above we note that
A × B = {(red,b), (red,c), (red,s), (blue,b), (blue,c), (blue,s)}.
Again, consider the two sets:
A = {DL, MP, KA}, where DL, MP, KA represent Delhi,
03}representing codes for the licence plates of vehicles issued
by DL, MP and KA .
were making codes for the licence plates of vehicles, with the
restriction that the code begins with an element from set A,
which are the pairs available from these sets and how many such
pairs will there be (Fig 2.2)?
The available pairs are:(DL,01), (DL,02), (DL,03), (MP,01), (MP,02), (MP,03),
(KA,01), (KA,02), (KA,03) and the product of set A and set B is given by
A × B = {(DL,01), (DL,02), (DL,03), (MP,01), (MP,02), (MP,03), (KA,01), (KA,02),
(KA,03)}.
It can easily be seen that there will be 9 such pairs in the Cartesian product, since
there are 3 elements in each of the sets A and B. This gives us 9 possible codes. Also
note that the order in which these elements are paired is crucial. For example, the code
(DL, 01) will not be the same as the code (01, DL).
As a final illustration, consider the two sets A= {a
1
, a
2
} and
B = {b
1
, b
2
, b
3
, b
4
} (Fig 2.3).
A × B = {( a
1
, b
1
), (a
1
, b
2
), (a
1
, b
3
), (a
1
, b
4
), (a
2
, b
1
), (a
2
, b
2
),
(a
2
, b
3
), (a
2
, b
4
)}.
The 8 ordered pairs thus formed can represent the position of points in
the plane if A and B are subsets of the set of real numbers and it is
obvious that the point in the position (a
1
, b
2
) will be distinct from the point
in the position (b
2
, a
1
).
Remarks
(i) Two ordered pairs are equal, if and only if  the corresponding first elements
are equal and the second  elements are also equal.
DL MP KA
03
02
01
Fig 2.2
Fig 2.3
Rationalised 2023-24
26 MATHEMATICS
(ii) If there are p elements in A and q elements in B, then there will be pq
elements in A × B, i.e.,  if n(A) = p and n(B) = q,  then n(A × B) = pq.
(iii) If A and B are non-empty sets and either A or B is an infinite set, then so is
A × B.
(iv) A × A × A = {(a, b, c) : a, b, c ? A}. Here (a, b, c) is called an ordered
triplet.
Example 1 If  (x + 1, y – 2) = (3,1), find the values of x and y.
Solution Since the ordered pairs are equal, the corresponding elements are equal.
Therefore x + 1 = 3  and y – 2 = 1.
Solving we get x = 2 and y = 3.
Example 2 If P = {a, b, c} and Q = {r}, form the sets P × Q and Q × P.
Are these two products equal?
Solution By the definition of the cartesian product,
P × Q =  {(a, r), (b, r), (c, r)} and Q × P =  {(r, a), (r, b), (r, c)}
Since, by the definition of equality of ordered pairs, the pair (a, r) is not equal to the pair
(r, a), we conclude that P × Q ? Q × P.
However, the number of elements in each set will be the same.
Example 3 Let A = {1,2,3}, B = {3,4} and C = {4,5,6}. Find
(i) A × (B n C) (ii) (A × B) n (A × C)
(iii) A × (B ? C) (iv) (A × B) ? (A × C)
Solution (i) By the definition of the intersection of two sets, (B n C) = {4}.
Therefore, A × (B n C) = {(1,4), (2,4), (3,4)}.
(ii) Now (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) n (A × C)  = {(1, 4), (2, 4), (3, 4)}.
(iii) Since, (B ? C) = {3, 4, 5, 6}, we have
A × (B ? 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)}.
(iv) Using the sets A × B and A × C from part (ii) above, we obtain
(A × B) ? (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)}.
Rationalised 2023-24
RELATIONS AND FUNCTIONS          27
Example 4 If P = {1, 2}, form the set P × P × P.
Solution We have,  P × P × P =  {(1,1,1), (1,1,2), (1,2,1), (1,2,2), (2,1,1), (2,1,2), (2,2,1),
(2,2,2)}.
Example 5 If R is the set of all real numbers, what do the cartesian products R × R
and R × R × R represent?
Solution The Cartesian product R × R represents the set R × R={(x, y) : x, y ? R}
which represents the coordinates of all the points in two dimensional space and the
cartesian product R × R × R represents the set R × R × R ={(x, y, z) : x, y, z ? R}
which  represents the coordinates of all the points in three-dimensional space.
Example 6 If A × B ={(p, q),(p, r), (m, q), (m, r)}, find A and B.
Solution A = set of first elements = {p, m}
B = set of second elements = {q, r}.
EXERCISE 2.1
1. If
2 5 1
1
3 3 3 3
x
,y – ,
? ? ? ?
+ =
? ? ? ?
? ? ? ?
, find the values of x and y.
2. If the set A has 3 elements and the set B = {3, 4, 5}, then find the number of
elements in (A×B).
3. If G = {7, 8} and H = {5, 4, 2}, find G × H and H × G.
4. 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 n f) = f.
5. If A = {–1, 1}, find A × A × A.
6. If A × B = {(a, x),(a , y), (b, x), (b, y)}. Find A and B.
7. Let A = {1, 2}, B = {1, 2, 3, 4}, C = {5, 6} and D = {5, 6, 7, 8}. Verify that
(i) A × (B n C) = (A × B) n (A × C). (ii) A × C is a subset of  B × D.
8. Let A = {1, 2} and B = {3, 4}.  Write A × B. How many subsets will A × B have?
List them.
9. 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.
Rationalised 2023-24
Page 5

vMathematics is the indispensable instrument of
all physical research. – BERTHELOT v
2.1  Introduction
Much of mathematics is about finding a pattern – a
recognisable link between quantities that change. In our
daily life, we come across many patterns that characterise
relations such as brother and sister, father and son, teacher
and student. In mathematics also, we come across many
relations such as number m is less than number n, line l is
parallel to line m, set A is a subset of set B. In all these, we
notice that a relation involves pairs of objects in certain
order. In this Chapter, we will learn how to link pairs of
objects from two sets and then introduce relations between
the two objects in the pair. Finally, we will learn about
special relations which will qualify to be functions. The
concept of function is very important in mathematics since it captures the idea of a
mathematically precise correspondence between one quantity with the other.
2.2  Cartesian Products of Sets
Suppose A is a set of 2 colours and B is a set of  3 objects, i.e.,
A = {red, blue}and B = {b, c, s},
where b, c and s represent a particular bag, coat and shirt, respectively.
How many pairs of coloured objects can be made from these two sets?
Proceeding in a very orderly manner, we can see that there will be 6
distinct pairs as given below:
(red, b), (red, c), (red, s), (blue, b), (blue, c), (blue, s).
Thus, we get 6 distinct objects (Fig 2.1).
Let us recall from our earlier classes that an ordered pair of elements
taken from any two sets P and Q is a pair of elements written in small
Fig 2.1
Chapter 2
RELATIONS AND FUNCTIONS
G . W.  Leibnitz
(1646–1716)
Rationalised 2023-24
RELATIONS AND FUNCTIONS          25
brackets and grouped together in a particular order, i.e., (p,q), p ? P and  q ? Q . This
Definition 1 Given two non-empty sets P and Q. The cartesian product P × Q is the
set of all ordered pairs of elements from P and Q, i.e.,
P × Q = { (p,q) : p  ? P, q  ? Q }
If either P or Q is the null set, then P × Q will also be empty set, i.e., P × Q = f
From the illustration given above we note that
A × B = {(red,b), (red,c), (red,s), (blue,b), (blue,c), (blue,s)}.
Again, consider the two sets:
A = {DL, MP, KA}, where DL, MP, KA represent Delhi,
03}representing codes for the licence plates of vehicles issued
by DL, MP and KA .
were making codes for the licence plates of vehicles, with the
restriction that the code begins with an element from set A,
which are the pairs available from these sets and how many such
pairs will there be (Fig 2.2)?
The available pairs are:(DL,01), (DL,02), (DL,03), (MP,01), (MP,02), (MP,03),
(KA,01), (KA,02), (KA,03) and the product of set A and set B is given by
A × B = {(DL,01), (DL,02), (DL,03), (MP,01), (MP,02), (MP,03), (KA,01), (KA,02),
(KA,03)}.
It can easily be seen that there will be 9 such pairs in the Cartesian product, since
there are 3 elements in each of the sets A and B. This gives us 9 possible codes. Also
note that the order in which these elements are paired is crucial. For example, the code
(DL, 01) will not be the same as the code (01, DL).
As a final illustration, consider the two sets A= {a
1
, a
2
} and
B = {b
1
, b
2
, b
3
, b
4
} (Fig 2.3).
A × B = {( a
1
, b
1
), (a
1
, b
2
), (a
1
, b
3
), (a
1
, b
4
), (a
2
, b
1
), (a
2
, b
2
),
(a
2
, b
3
), (a
2
, b
4
)}.
The 8 ordered pairs thus formed can represent the position of points in
the plane if A and B are subsets of the set of real numbers and it is
obvious that the point in the position (a
1
, b
2
) will be distinct from the point
in the position (b
2
, a
1
).
Remarks
(i) Two ordered pairs are equal, if and only if  the corresponding first elements
are equal and the second  elements are also equal.
DL MP KA
03
02
01
Fig 2.2
Fig 2.3
Rationalised 2023-24
26 MATHEMATICS
(ii) If there are p elements in A and q elements in B, then there will be pq
elements in A × B, i.e.,  if n(A) = p and n(B) = q,  then n(A × B) = pq.
(iii) If A and B are non-empty sets and either A or B is an infinite set, then so is
A × B.
(iv) A × A × A = {(a, b, c) : a, b, c ? A}. Here (a, b, c) is called an ordered
triplet.
Example 1 If  (x + 1, y – 2) = (3,1), find the values of x and y.
Solution Since the ordered pairs are equal, the corresponding elements are equal.
Therefore x + 1 = 3  and y – 2 = 1.
Solving we get x = 2 and y = 3.
Example 2 If P = {a, b, c} and Q = {r}, form the sets P × Q and Q × P.
Are these two products equal?
Solution By the definition of the cartesian product,
P × Q =  {(a, r), (b, r), (c, r)} and Q × P =  {(r, a), (r, b), (r, c)}
Since, by the definition of equality of ordered pairs, the pair (a, r) is not equal to the pair
(r, a), we conclude that P × Q ? Q × P.
However, the number of elements in each set will be the same.
Example 3 Let A = {1,2,3}, B = {3,4} and C = {4,5,6}. Find
(i) A × (B n C) (ii) (A × B) n (A × C)
(iii) A × (B ? C) (iv) (A × B) ? (A × C)
Solution (i) By the definition of the intersection of two sets, (B n C) = {4}.
Therefore, A × (B n C) = {(1,4), (2,4), (3,4)}.
(ii) Now (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) n (A × C)  = {(1, 4), (2, 4), (3, 4)}.
(iii) Since, (B ? C) = {3, 4, 5, 6}, we have
A × (B ? 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)}.
(iv) Using the sets A × B and A × C from part (ii) above, we obtain
(A × B) ? (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)}.
Rationalised 2023-24
RELATIONS AND FUNCTIONS          27
Example 4 If P = {1, 2}, form the set P × P × P.
Solution We have,  P × P × P =  {(1,1,1), (1,1,2), (1,2,1), (1,2,2), (2,1,1), (2,1,2), (2,2,1),
(2,2,2)}.
Example 5 If R is the set of all real numbers, what do the cartesian products R × R
and R × R × R represent?
Solution The Cartesian product R × R represents the set R × R={(x, y) : x, y ? R}
which represents the coordinates of all the points in two dimensional space and the
cartesian product R × R × R represents the set R × R × R ={(x, y, z) : x, y, z ? R}
which  represents the coordinates of all the points in three-dimensional space.
Example 6 If A × B ={(p, q),(p, r), (m, q), (m, r)}, find A and B.
Solution A = set of first elements = {p, m}
B = set of second elements = {q, r}.
EXERCISE 2.1
1. If
2 5 1
1
3 3 3 3
x
,y – ,
? ? ? ?
+ =
? ? ? ?
? ? ? ?
, find the values of x and y.
2. If the set A has 3 elements and the set B = {3, 4, 5}, then find the number of
elements in (A×B).
3. If G = {7, 8} and H = {5, 4, 2}, find G × H and H × G.
4. 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 n f) = f.
5. If A = {–1, 1}, find A × A × A.
6. If A × B = {(a, x),(a , y), (b, x), (b, y)}. Find A and B.
7. Let A = {1, 2}, B = {1, 2, 3, 4}, C = {5, 6} and D = {5, 6, 7, 8}. Verify that
(i) A × (B n C) = (A × B) n (A × C). (ii) A × C is a subset of  B × D.
8. Let A = {1, 2} and B = {3, 4}.  Write A × B. How many subsets will A × B have?
List them.
9. 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.
Rationalised 2023-24
28 MATHEMATICS
10. 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.
2.3  Relations
Consider the two sets P = {a, b, c} and Q = {Ali, Bhanu, Binoy, Chandra, Divya}.
The cartesian product of
P and Q has 15 ordered pairs which
can be listed as P × Q = {(a, Ali),
(a,Bhanu), (a, Binoy), ..., (c, Divya)}.
We can now obtain a subset of
P × Q by introducing a relation R
between the first element x and the
second element y of each ordered pair
(x, y) as
R= { (x,y): x is the first letter of the name y, x ? P, y ? Q}.
Then R = {(a, Ali), (b, Bhanu), (b, Binoy), (c, Chandra)}
A visual representation of this relation R (called an arrow diagram) is shown
in Fig 2.4.
Definition 2 A relation R from a non-empty set A to a non-empty set B is a subset of
the cartesian product  A × B. The subset is derived by describing a relationship between
the first element and the second element of the ordered pairs in A × B. The second
element is called the image of  the first element.
Definition 3 The set of all first elements of the ordered pairs in a relation R from a set
A to a set B is called the domain of the relation R.
Definition 4 The set of all second elements in a relation R from a set A to a set B is
called the range of the relation R. The whole set B is called the codomain of the
relation R. Note that range ? codomain.
Remarks (i) A relation may be represented algebraically either by the Roster
method or by the Set-builder method.
(ii) An arrow diagram is a visual representation of a relation.
Example 7 Let A = {1, 2, 3, 4, 5, 6}. Define a relation R from A to A by
R = {(x, y) : y =  x + 1 }
(i) Depict this relation using an arrow diagram.
(ii) Write down the domain, codomain and range of R.
Solution (i) By the definition of the relation,
R = {(1,2), (2,3), (3,4), (4,5), (5,6)}.
Fig 2.4
Rationalised 2023-24
```

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## FAQs on NCERT Textbook - Relations and Functions - NCERT Textbooks (Class 6 to Class 12) - CTET & State TET

 1. What is the difference between relations and functions?
Ans. Relations and functions are both concepts in mathematics, but they have some differences. A relation is a set of ordered pairs, where each pair consists of an input value and an output value. On the other hand, a function is a special kind of relation where each input value is associated with exactly one output value. In other words, a function is a relation that passes the vertical line test, meaning that no two distinct input values can have the same output value.
 2. How can we determine if a relation is a function or not?
Ans. To determine if a relation is a function or not, we can use the vertical line test. If any vertical line intersects the graph of the relation at more than one point, then the relation is not a function. This is because a function should have only one output value for each input value. If the relation passes the vertical line test, then it is a function.
 3. What are the different types of functions?
Ans. There are several types of functions based on their properties and characteristics. Some common types of functions include: 1. One-to-one function: A function where each input value is associated with a unique output value, and no two distinct input values have the same output value. 2. Onto function: A function where each output value is associated with at least one input value. In other words, all the elements of the codomain are used. 3. Constant function: A function where the output value is the same for every input value. It represents a horizontal line on a graph. 4. Linear function: A function that can be represented by a straight line on a graph, where the output value is directly proportional to the input value. 5. Quadratic function: A function that can be represented by a parabola on a graph, where the output value is a quadratic expression of the input value.
 4. How can we determine the domain and range of a function?
Ans. The domain of a function is the set of all possible input values for which the function is defined. To determine the domain, we need to consider any restrictions or limitations on the input values. For example, if a function contains a square root, the input values must be non-negative, so the domain would be all real numbers greater than or equal to zero. The range of a function is the set of all possible output values that the function can produce. To determine the range, we can examine the graph of the function or analyze its behavior. For example, if a function is a quadratic function with a vertex at the minimum point, the range would be all real numbers greater than or equal to the y-coordinate of the vertex.
 5. How can we determine if two functions are equal?
Ans. Two functions are considered equal if they have the same domain, the same range, and they produce the same output value for every input value in their common domain. In other words, if the output value of one function matches the output value of the other function for every input value, then the two functions are equal. It is important to note that functions can have different equations or representations but still be considered equal as long as their outputs are the same for every input.

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