UNIT – I: RELATIONS AND FUNCTIONS
CHAPTER-1
RELATIONS AND FUNCTIONS
Relations
Topic-1
Concepts Covered Types of relations and their identification Equivalence class
Revision Notes
1. Definition
A relation R, from a non-empty set A to another non-empty set B is mathematically as an subset of A × B.
Equivalently, any subset of A × B is a relation from A to B. Thus, R is a relation from A to B
ÛRÍA×B
Û R Í {(a, b) : a Î A, b Î B}
Illustrations:
Let A = {1, 2, 4}, B = {4, 6}. Let R = {(1, 4), (1, 6), (2, 4), (2, 6), (4, 4) (4, 6)}. Here R Í A × B and therefore R is a
(a)
relation from A to B.
(b) Let A = {1, 2, 3}, B = {2, 3, 5, 7}, Let R = {(2, 3), (3, 5), (5, 7)}. Here R Ë A × B and therefore R is not a relation
from A to B. Since (5, 7) Î R but (5, 7) Ï A × B.
(c) Let A = {–1, 1, 2}, B = {1, 4, 9, 10} let a Î A and b Î B and a R b means a2 = b then, R = {(–1, 1), (1, 1), (2, 4)}.
Note:
A relation from A to B is also called a relation from A into B.
(a, b) Î R is also written as aRb (read as a is related to b).
Let A and B be two non-empty finite sets having p and q elements respectively. Then n(A × B) = n(A)·n(B) = pq.
Then total number of subsets of A × B = 2pq. Since each subset of A × B is a relation from A to B, therefore total
number of relations from A to B will be 2pq.
2. Domain & range of a relation
(a) Domain of a relation: Let R be a relation from A to B. The domain of relation R is the set of all those elements
a Î A such that (a, b) Î R " b Î B.
Thus, Dom.(R) ={a Î A : (a, b) Î R " b Î B}.
,2 Oswaal CBSE Revision Notes Chapterwise & Topicwise, MATHEMATICS, Class-XII
That is, the domain of R is the set of first components of all the ordered pairs which
belong to R.
(b) Range of a relation: Let R be a relation from A to B. The range of relation R is the set of all those elements
b Î B such that (a, b) Î R " a Î A.
Thus, Range of R ={b Î B : (a, b) Î R " a Î A}.
That is, the range of R is the set of second components of all the ordered pairs which
belong to R.
(c) Co-domain of a relation: Let R be a relation from A to B. Then B is called the co-domain of the relation R.
So we can observe that co-domain of a relation R from A into B is the set B as a whole.
Illustrations: Let a Î A and b Î B and
(i)
Let A = {1, 2, 3, 7}, B
= {3, 6}. If aRb means a < b.
Then we have
R = {(1, 3), (1, 6), (2, 3), (2, 6), (3, 6)}.
Here, Dom.(R) = {1, 2, 3},
Range of R= {3, 6}, Co-domain of R = B = {3, 6}
(ii) Let A = {1, 2, 3}, B = {2, 4, 6, 8}.
If R1 = {(1, 2), (2, 4), (3, 6)},
and R2 = {(2, 4}, (2, 6), (3, 8), (1, 6)}
Then both R1 and R2 are related from A to B because
R1 Í A × B, R2 Í A × B
Here, Dom
(R1) = {1, 2, 3}, Range of R1 = {2, 4, 6};
Dom(R2) = {2, 3, 1}, Range of R2 = {4, 6, 8}
3. Types of relations from one set to another set
(a) Empty relation: A relation R from A to B is called an empty relation or a void relation from A to B if
R = f.
For example, Let
A = {2, 4, 6}, B = {7, 11}
Let R = {(a, b) : a Î A, b Î B and |a – b| is even}.
Here R is an empty relation.
(b) Universal relation: A relation R from A to B is said to be the universal relation if R = A × B.
For example, Let
A = {1, 2}, B = {1, 3}
Let R = {(1, 1), (1, 3), (2, 1), (2, 3)}.
Here, R = A × B, so relation R is a universal relation.
Note:
The void relation i.e., f and universal relation i.e., A × A on A are respectively the smallest and largest relations
defined on the set A. Also these are also called Trivial Relations and other relation is called a Non-Trivial Relation.
The relations R = f and R = A × A are two extreme relations.
(c) Identity relation: A relation R defined on a set A is said to be the identity relation on A if R = {(a, b) : a Î A, b Î A
and a = b}
Thus identity relation
R = {(a, a) : " a Î A}
, Oswaal CBSE Revision Notes Chapterwise & Topicwise, MATHEMATICS, Class-XII 3
The identity relation on set A is also denoted by IA.
For example, Let A = {1, 2, 3, 4},
Then IA = {(1, 1), (2, 2), (3, 3), (4, 4)}.
But the relation given by
R = {(1, 1), (2, 2), (1, 3), (4, 4)}
is not an identity relation because element of IA is not related to elements 1 and 3.
Note:
In an identity relation on A every element of A should be related to itself only.
(d) Reflexive relation: A relation R defined on a set A is said to be reflexive if a R a " a Î A i.e., (a, a) Î
R " a Î A.
For example, Let A ={1, 2, 3} and R1, R2, R3 be the relations given as
R1 = {(1, 1), (2, 2), (3, 3)},
R2 = {(1, 1), (2, 2), (3, 3), (1, 2), (2, 1), (1, 3)} and
R3 = {(2, 2), (2, 3), (3, 2), (1, 1)}
Here R1 and R2 are reflexive relations on A but R3 is not reflexive as 3 Î A but (3, 3) Ï R3.
Note:
The universal relation on a non-void set A is reflexive.
The identity relation is always a reflexive relation but the converse may or may not be true. As shown in the
example above, R1 is both identity as well as reflexive relation on A but R2 is only reflexive relation on A.
(e) Symmetric relation: A relation R defined on a set A is symmetric if
(a, b) Î R Þ (b, a) Î R " a, b Î A i.e., aRb Þ bRa (i.e., whenever aRb then bRa).
For example, Let A = {1, 2, 3},
R1 = {(1, 2), (2, 1)}, R2 = {(1, 2), (2, 1), (1, 3), (3, 1)}.
R3 = {(1, 2), (2, 1), (1, 3), (3, 1), (2, 3), (3, 2)}
R4 = {(1, 3), (3, 1), (2, 3)}
Here R1, R2 and R3 are symmetric relations on A. But R4 is not symmetric because (2, 3) Î R4 but (3, 2) Ï R4.
(f) Transitive relation: A relation R on a set A is transitive if (a, b) Î R and (b, c) Î R Þ (a, c) Î R
i.e., aRb and bRc Þ aRc.
For example, Let A = {1, 2, 3},
R1 = {(1, 2), (2, 3), (1, 3), (3, 2)}
and R2 = {(1, 3), (3, 2), (1, 2)}
Here R2 is transitive relation whereas R1 is not transitive because (2, 3) Î R1 and (3, 2) Î R1 but (2, 2) Ï R1.
(g) Equivalence relation: Let A be a non-empty set, then a relation R on A is said to be an equivalence
relation if
(i) R is reflexive i.e.,
(a, a) Î R " a Î A i.e., aRa.
(ii)R is symmetric i.e.,
(a, b) Î R
Þ (b, a) Î R " a, b Î A i.e., aRb Þ bRa.
(iii)R is transitive i.e.,
(a, b) Î R and (b, c) Î R
Þ (a, c) Î R " a, b, c Î A
i.e., aRb and bRc Þ aRc.
For example, Let A = {1, 2, 3}
R = {(1, 2), (1, 1), (2, 1), (2, 2), (3, 3) (1, 3), (3, 1), (3, 2), (2, 3)}
Here R is reflexive, symmetric and transitive. So R is an equivalence relation on A.
Equivalence classes: Let A be an equivalence relation in a set A and let a Î A. Then, the set of all those elements
of A which are related to a, is called equivalence class determined by a and it is denoted by [a]. Thus, [a] = {b Î A :
(a, b) Î A}
Mnemonics
Types of relation Strawberry – Symmetric
RIPE STRAWBERRY TO EAT To – transitive
Interpretations Eat - Equivalence
Ripe – reflexive
, 4 Oswaal CBSE Revision Notes Chapterwise & Topicwise, MATHEMATICS, Class-XII
Note:
Two equivalence classes are either disjoint or identical.
An equivalence relation R on a set A partitions the set into mutually disjoint equivalence classes.
An important property of an equivalence relation is that it divides the set into pair-wise disjoint subsets called
equivalence classes whose collection is called a partition of the set.
Note that the union of all equivalence classes give the whole set.
e.g., Let R denotes the equivalence relation in the set Z of integers given by R = {(a, b) : 2 divides a – b}. Then the
equivalence class [0] is [0] = [0, ± 2, ± 4, ± 6,.....].
Key Words
Disjoint: These are sets which have no elements in common.
4. Tabular representation of a relation
In this form of representation of a relation R from set A to set B, elements of A and B are written in the first column
and first row respectively. If (a, b) Î R then we write ‘1‘ in the row containing a and column containing b and
if (a, b) Ï R then we write ‘0‘ in the same manner.
For example, Let A = {1, 2, 3},
B = {2, 5} and R = {(1, 2), (2, 5), (3, 2)}, then
R 2 5
1 1 0
2 0 1
3 1 0
5. Inverse relation
Let R Í A × B be a relation from A to B. Then, the inverse relation of R, to be denoted by R–1, is a relation
from B to A defined by R–1 = {(b, a) : (a, b) Î R}
Thus (a, b) Î R Û (b, a) Î R–1 " a Î A, b Î B.
Clearly, Domain (R–1) = Range of R, Range of R–1 = Domain (R).
Key Words
Domain and Range: The set of x coordinate values is called domain and the set of y coordinate values is called
range.
Also, (R–1)–1 = R.
For example, Let A = {1, 2, 4}, B ={3, 0} and let R = {(1, 3), (4, 0), (2, 3)} be a relation from A to B, then
R–1 = {(3, 1), (0, 4), (3, 2)}.
Key Facts
1. (i) A relation R from A to B is an empty relation or void relation if R = f
(ii) A relation R on a set A is an empty relation or void relation if R = f
2. (i) A relation R from A to B is a universal relation if R = A × B.
(ii) A relation R on a set A is an universal relation if R = A × A.
3. A relation R on a set A is reflexive if aRa, " a Î A.
4. A relation R on a set A is symmetric if whenever aRb, then bRa for all a, b Î A.
5. A relation R on a set A is transitive if whenever aRb and bRc then aRc for all a, b, c Î A.
6. A relation R on A is identity relation if R = {(a, a) " a Î A} i.e., R contains only elements of the type (a, a) " a Î A and it
contains no other element.
7.
A relation R on a non-empty set A is an equivalence relation if the following conditions are satisfied :
(i) R is reflexive i.e., for every a Î A, (a, a) Î R i.e., aRa.
(ii) R is symmetric i.e., for a, b Î A, aRb Þ bRa i.e., (a, b) Î R Þ (b, a) Î R.
(iii) R is transitive i.e., for all a, b, c Î A, we have, aRb and bRc Þ aRc i.e., (a, b) Î R and (b, c) Î R Þ (a, c) Î R.
CHAPTER-1
RELATIONS AND FUNCTIONS
Relations
Topic-1
Concepts Covered Types of relations and their identification Equivalence class
Revision Notes
1. Definition
A relation R, from a non-empty set A to another non-empty set B is mathematically as an subset of A × B.
Equivalently, any subset of A × B is a relation from A to B. Thus, R is a relation from A to B
ÛRÍA×B
Û R Í {(a, b) : a Î A, b Î B}
Illustrations:
Let A = {1, 2, 4}, B = {4, 6}. Let R = {(1, 4), (1, 6), (2, 4), (2, 6), (4, 4) (4, 6)}. Here R Í A × B and therefore R is a
(a)
relation from A to B.
(b) Let A = {1, 2, 3}, B = {2, 3, 5, 7}, Let R = {(2, 3), (3, 5), (5, 7)}. Here R Ë A × B and therefore R is not a relation
from A to B. Since (5, 7) Î R but (5, 7) Ï A × B.
(c) Let A = {–1, 1, 2}, B = {1, 4, 9, 10} let a Î A and b Î B and a R b means a2 = b then, R = {(–1, 1), (1, 1), (2, 4)}.
Note:
A relation from A to B is also called a relation from A into B.
(a, b) Î R is also written as aRb (read as a is related to b).
Let A and B be two non-empty finite sets having p and q elements respectively. Then n(A × B) = n(A)·n(B) = pq.
Then total number of subsets of A × B = 2pq. Since each subset of A × B is a relation from A to B, therefore total
number of relations from A to B will be 2pq.
2. Domain & range of a relation
(a) Domain of a relation: Let R be a relation from A to B. The domain of relation R is the set of all those elements
a Î A such that (a, b) Î R " b Î B.
Thus, Dom.(R) ={a Î A : (a, b) Î R " b Î B}.
,2 Oswaal CBSE Revision Notes Chapterwise & Topicwise, MATHEMATICS, Class-XII
That is, the domain of R is the set of first components of all the ordered pairs which
belong to R.
(b) Range of a relation: Let R be a relation from A to B. The range of relation R is the set of all those elements
b Î B such that (a, b) Î R " a Î A.
Thus, Range of R ={b Î B : (a, b) Î R " a Î A}.
That is, the range of R is the set of second components of all the ordered pairs which
belong to R.
(c) Co-domain of a relation: Let R be a relation from A to B. Then B is called the co-domain of the relation R.
So we can observe that co-domain of a relation R from A into B is the set B as a whole.
Illustrations: Let a Î A and b Î B and
(i)
Let A = {1, 2, 3, 7}, B
= {3, 6}. If aRb means a < b.
Then we have
R = {(1, 3), (1, 6), (2, 3), (2, 6), (3, 6)}.
Here, Dom.(R) = {1, 2, 3},
Range of R= {3, 6}, Co-domain of R = B = {3, 6}
(ii) Let A = {1, 2, 3}, B = {2, 4, 6, 8}.
If R1 = {(1, 2), (2, 4), (3, 6)},
and R2 = {(2, 4}, (2, 6), (3, 8), (1, 6)}
Then both R1 and R2 are related from A to B because
R1 Í A × B, R2 Í A × B
Here, Dom
(R1) = {1, 2, 3}, Range of R1 = {2, 4, 6};
Dom(R2) = {2, 3, 1}, Range of R2 = {4, 6, 8}
3. Types of relations from one set to another set
(a) Empty relation: A relation R from A to B is called an empty relation or a void relation from A to B if
R = f.
For example, Let
A = {2, 4, 6}, B = {7, 11}
Let R = {(a, b) : a Î A, b Î B and |a – b| is even}.
Here R is an empty relation.
(b) Universal relation: A relation R from A to B is said to be the universal relation if R = A × B.
For example, Let
A = {1, 2}, B = {1, 3}
Let R = {(1, 1), (1, 3), (2, 1), (2, 3)}.
Here, R = A × B, so relation R is a universal relation.
Note:
The void relation i.e., f and universal relation i.e., A × A on A are respectively the smallest and largest relations
defined on the set A. Also these are also called Trivial Relations and other relation is called a Non-Trivial Relation.
The relations R = f and R = A × A are two extreme relations.
(c) Identity relation: A relation R defined on a set A is said to be the identity relation on A if R = {(a, b) : a Î A, b Î A
and a = b}
Thus identity relation
R = {(a, a) : " a Î A}
, Oswaal CBSE Revision Notes Chapterwise & Topicwise, MATHEMATICS, Class-XII 3
The identity relation on set A is also denoted by IA.
For example, Let A = {1, 2, 3, 4},
Then IA = {(1, 1), (2, 2), (3, 3), (4, 4)}.
But the relation given by
R = {(1, 1), (2, 2), (1, 3), (4, 4)}
is not an identity relation because element of IA is not related to elements 1 and 3.
Note:
In an identity relation on A every element of A should be related to itself only.
(d) Reflexive relation: A relation R defined on a set A is said to be reflexive if a R a " a Î A i.e., (a, a) Î
R " a Î A.
For example, Let A ={1, 2, 3} and R1, R2, R3 be the relations given as
R1 = {(1, 1), (2, 2), (3, 3)},
R2 = {(1, 1), (2, 2), (3, 3), (1, 2), (2, 1), (1, 3)} and
R3 = {(2, 2), (2, 3), (3, 2), (1, 1)}
Here R1 and R2 are reflexive relations on A but R3 is not reflexive as 3 Î A but (3, 3) Ï R3.
Note:
The universal relation on a non-void set A is reflexive.
The identity relation is always a reflexive relation but the converse may or may not be true. As shown in the
example above, R1 is both identity as well as reflexive relation on A but R2 is only reflexive relation on A.
(e) Symmetric relation: A relation R defined on a set A is symmetric if
(a, b) Î R Þ (b, a) Î R " a, b Î A i.e., aRb Þ bRa (i.e., whenever aRb then bRa).
For example, Let A = {1, 2, 3},
R1 = {(1, 2), (2, 1)}, R2 = {(1, 2), (2, 1), (1, 3), (3, 1)}.
R3 = {(1, 2), (2, 1), (1, 3), (3, 1), (2, 3), (3, 2)}
R4 = {(1, 3), (3, 1), (2, 3)}
Here R1, R2 and R3 are symmetric relations on A. But R4 is not symmetric because (2, 3) Î R4 but (3, 2) Ï R4.
(f) Transitive relation: A relation R on a set A is transitive if (a, b) Î R and (b, c) Î R Þ (a, c) Î R
i.e., aRb and bRc Þ aRc.
For example, Let A = {1, 2, 3},
R1 = {(1, 2), (2, 3), (1, 3), (3, 2)}
and R2 = {(1, 3), (3, 2), (1, 2)}
Here R2 is transitive relation whereas R1 is not transitive because (2, 3) Î R1 and (3, 2) Î R1 but (2, 2) Ï R1.
(g) Equivalence relation: Let A be a non-empty set, then a relation R on A is said to be an equivalence
relation if
(i) R is reflexive i.e.,
(a, a) Î R " a Î A i.e., aRa.
(ii)R is symmetric i.e.,
(a, b) Î R
Þ (b, a) Î R " a, b Î A i.e., aRb Þ bRa.
(iii)R is transitive i.e.,
(a, b) Î R and (b, c) Î R
Þ (a, c) Î R " a, b, c Î A
i.e., aRb and bRc Þ aRc.
For example, Let A = {1, 2, 3}
R = {(1, 2), (1, 1), (2, 1), (2, 2), (3, 3) (1, 3), (3, 1), (3, 2), (2, 3)}
Here R is reflexive, symmetric and transitive. So R is an equivalence relation on A.
Equivalence classes: Let A be an equivalence relation in a set A and let a Î A. Then, the set of all those elements
of A which are related to a, is called equivalence class determined by a and it is denoted by [a]. Thus, [a] = {b Î A :
(a, b) Î A}
Mnemonics
Types of relation Strawberry – Symmetric
RIPE STRAWBERRY TO EAT To – transitive
Interpretations Eat - Equivalence
Ripe – reflexive
, 4 Oswaal CBSE Revision Notes Chapterwise & Topicwise, MATHEMATICS, Class-XII
Note:
Two equivalence classes are either disjoint or identical.
An equivalence relation R on a set A partitions the set into mutually disjoint equivalence classes.
An important property of an equivalence relation is that it divides the set into pair-wise disjoint subsets called
equivalence classes whose collection is called a partition of the set.
Note that the union of all equivalence classes give the whole set.
e.g., Let R denotes the equivalence relation in the set Z of integers given by R = {(a, b) : 2 divides a – b}. Then the
equivalence class [0] is [0] = [0, ± 2, ± 4, ± 6,.....].
Key Words
Disjoint: These are sets which have no elements in common.
4. Tabular representation of a relation
In this form of representation of a relation R from set A to set B, elements of A and B are written in the first column
and first row respectively. If (a, b) Î R then we write ‘1‘ in the row containing a and column containing b and
if (a, b) Ï R then we write ‘0‘ in the same manner.
For example, Let A = {1, 2, 3},
B = {2, 5} and R = {(1, 2), (2, 5), (3, 2)}, then
R 2 5
1 1 0
2 0 1
3 1 0
5. Inverse relation
Let R Í A × B be a relation from A to B. Then, the inverse relation of R, to be denoted by R–1, is a relation
from B to A defined by R–1 = {(b, a) : (a, b) Î R}
Thus (a, b) Î R Û (b, a) Î R–1 " a Î A, b Î B.
Clearly, Domain (R–1) = Range of R, Range of R–1 = Domain (R).
Key Words
Domain and Range: The set of x coordinate values is called domain and the set of y coordinate values is called
range.
Also, (R–1)–1 = R.
For example, Let A = {1, 2, 4}, B ={3, 0} and let R = {(1, 3), (4, 0), (2, 3)} be a relation from A to B, then
R–1 = {(3, 1), (0, 4), (3, 2)}.
Key Facts
1. (i) A relation R from A to B is an empty relation or void relation if R = f
(ii) A relation R on a set A is an empty relation or void relation if R = f
2. (i) A relation R from A to B is a universal relation if R = A × B.
(ii) A relation R on a set A is an universal relation if R = A × A.
3. A relation R on a set A is reflexive if aRa, " a Î A.
4. A relation R on a set A is symmetric if whenever aRb, then bRa for all a, b Î A.
5. A relation R on a set A is transitive if whenever aRb and bRc then aRc for all a, b, c Î A.
6. A relation R on A is identity relation if R = {(a, a) " a Î A} i.e., R contains only elements of the type (a, a) " a Î A and it
contains no other element.
7.
A relation R on a non-empty set A is an equivalence relation if the following conditions are satisfied :
(i) R is reflexive i.e., for every a Î A, (a, a) Î R i.e., aRa.
(ii) R is symmetric i.e., for a, b Î A, aRb Þ bRa i.e., (a, b) Î R Þ (b, a) Î R.
(iii) R is transitive i.e., for all a, b, c Î A, we have, aRb and bRc Þ aRc i.e., (a, b) Î R and (b, c) Î R Þ (a, c) Î R.
