RELATIONS

CARTESIAN PRODUCT OF SETS

Let A and B be two non – empty sets . The set of all ordered pairs (a , b ) such that a ∈A and b ∈ B is called the Cartesian product of set A with set B and is denoted by A×B .
Thus ,
A×B = { ( a ,b):a ∈A and b ∈B }

RELATION

Let A and B be two sets . Then a relation R from a set A to a set B is a subset of A×B .
Thus , R is a relation from A to B ↔ R ⊆A × B .
If R is a relation from a non – void set A to a non – void set B and if (a, b) ∈R , then we write a R b which
is read as ‘a is related to b by the relation R ‘.

Note –
If A and B are finite sets consisting of m and n elements respectively , then total number of relations
from A to B is 2^mn .

DOMAIN –
Let R be a relation from a set A to a set B . Then the set of all first components or coordinates of the
ordered pairs belonging to R is called domain of R .
Thus , domain of R = { a : (a, b) ∈R }

RANGE –
Let R be a relation from a set A to a set B . Then the set of all second components or coordinates of the
ordered pairs belonging to R is called the range of R .
Thus , range of R = { b : (a , b ) ∈ R }

RELATION ON A SET –
Let A be a non – empty set . Then a relation from A to itself is called a relation on set A .

TYPES OF RELATIONS –

1.VOID RELATION –

Let A be a set . Then , ∅ ⊆ A×A and so it is a relation on A . This relation is called the void or empty
relation on a set A .

UNIVERSAL RELATION –
Let A be a set . Then , A×A ⊆ A× A and so it is a relation on A . This relation is called the universal
relation on A .
In other words , a relation R on a set is called a universal relation, if each element of A is related to every
element of A .
NOTE –
THE VOID RELATION AND THE UNIVERSAL RELATION ON A SET ARE RESPECTIVELY THE SMALLEST AND
THE LARGEST RELATION ON A SET .
IDENTITY RELATION –
Let A be a set . Then the relation ,
I = { (a , a) : a ∈ A } on A is called the identity relation on A .
In other words , a relation I on A is called the identity relation if every element of A is related to itself only.

REFLEXIVE RELATION –
A relation R on a set A is said to be reflexive if every element of A is related to itself .
Thus , R is reflexive ↔ (a , a) ∈ R for all a ∈ A .
NOTE – The universal relation on a non – empty set A is reflexive .

SYMMETRIC RELATION –
A relation R on a set A is said to be a symmetric relation if ( a , b ) ∈ R ⇒ ( b , a ) ∈ R for all a , b ∈ A .
Note –
The identity relation and universal relations on a non – void set are symmetric relations .

TRANSITIVE RELATION –
Let A be any set . A relation R on A is said to be a transitive relation if and only if (a , b ) ∈ R and ( b , c ) ∈ R
⇒ ( a , c ) ∈ R for all a , b ,c ∈ A .
Note –
The identity and universal relations on a non – void set are transitive .
ANTISYMMETRIC RELATION –
Let A be any set . A relation R on a set A is said to be an antisymmetric relation if and only if
(a , b ) ∈ R and ( b , a ) ∈ R ⇒ a = b for all a , b ∈ A .
Note – The identity relation on a non – void set is an antisymmetric relation.

EQUIVALENCE RELATION –
A relation R on a set A is said to be an equivalence relation on A if and only if
1.it is reflexive
2.it is symmetric
3.it is transitive .

SOME RESULTS ON RELATIONS –
(1)If R and S are two equivalence relation on a set A , then R ∩ S is also an equivalence relation.
(2)The union of two equivalence relations on a set is not necessarily an equivalence relation on the set .
(3)The inverse of an equivalence relation is an equivalence relation.
ILLUSTRATIVE EXAMPLES ---
(1)If A = { 1, 3,5 ,7 } , B = { 2, 4 , 6, 8, 10 } and R = { (1, 8 ) , ( 3 , 6) , (5, 2 ) , (1, 4) } is a relation from A to B ,
then
Domain (R) = { 1, 3, 5 }
Range (R ) = { 8, 6, 2 , 4 }
(2)Consider the relation R on the set A = { 1, 2, 3, 4, 5 } defined by
R = { (a, b ) : a – b = 12 }
Thus , a – b ≠ 12 for any two elements of A .
Therefore , (a , b ) ∉ R for any a , b ∈ A .
⇒ R does not contain any element of A ×A .
⇒ R is empty set .
⇒ R is the void relation on A .
(3)Consider the relation R on the set A = { 1, 2, 3, 4, 5, 6 } defined by
R = { (a, b ) : | a – b | ≥ 0 }
We observe that ,
|a – b | ≥ 0 for all a, b ∈ A .
⇒ (a , b ) ∈ R for all ( a, b ) ∈ A ×A .
⇒ each element of set A is related to every element of set A .
⇒ R = A × A
⇒R is an universal relation on set A .
(4)The relation ≤ ( less than or equal to ) on a set R of real numbers is antisymmetric relation
because
a ≤ b and b ≤ a ⇒ a = b for all a , b ∈ R .
(5)Let A = { a , b , c } .
Let R = { (a , b ) , ( b, a) ,( a, c) , (c, a) } be a relation defined on set A , then
(A)Reflexive –
Since , (a ,a) , (b, b) and (c, c) are not in R , so it is not a reflexive relation on A .
(B)Symmetric –
The ordered pairs obtained by interchanging the components of ordered pairs in R are also in R
.So ,R is a symmetric relation on set A .
(C)Transitive –
Clearly , (a, b ) ∈ R and (b, a) ∈ R but (a , a) ∉ R . So , it is not a transitive relation on set A .