Relations and Functions
Handout for module -1
In this module we will study about
 What is relation
 What are the different types of relations
1. Empty relation
2. Universal relation
3. Reflexive relation
4. Symmetric relation
5. Transitive relation
6. Equivalence relation
What is relation?
 Any Subset of the Cartesian product AxB is called a relation from
A to B
 Relation is collection of ordered pair (a,b) in which the two elements are
related by pre-defined ‘relation’.
 Relation between two elements is expressed as a formula or as a
statement
 If (a,b) ϵ R , we say a is related to b or we write it as aRb
 A relation on a set A means it is a subset of AxA or it is a relation from A
to A
Example 1
Consider a relation R defined on a set A ={ 1,2,3,4}. defined as a R b if a > b .
The set of elements in R = {(2,1),(3,1),(3,2 ),(4,1),(4,2),(4,3)}
Here every first element of the ordered pair is greater than second element
Example 2
A relation R in the set of integers is defined as a R b if 2 divides a-b
R={(0,0),(0,2),(0,4)....,(1,1),(1,3),(1,5)......,(2,0),(2,2),(2,4),....,....(0,-2)
(0,-4)....}
Here, 0 - 0 =0 is divisible by 2
0 – 2 = -2 is divisible by 2
............
...........
1. Empty Relation:
A relation R on a set A is called empty relation if no element of A is
related to any element of A.
ie, R = ɸ ⸦ A x A
2. Universal Relation:
A relation R in a set A is said to be universal relation , if each element
of A is related to every element of A.
ie, R = AxA
3. Reflexive Relation:
A relation R on a set A is said to be reflexive if every element of A is related to
itself
ie, if R is reflexive then (a , a) ϵ R for all a ϵ A
That is for a relation to be reflexive, the above condition is true for all elements
a in A
Working rule for reflexive
(i) For a finite set A, to say a relation is not reflexive it is enough to show
that there exist at least one element ‘a’ in A for which (a,a) does not exist
in R
ii)
For an infinite set A ,take an element aRb in R and replace b by a in
the definition of relation and check it is valid or not.
If valid then reflexive. If not valid then not reflexive.
Example-1
Consider the relation from Example 1 above, For the set A = {1,2,3,4}, relation
R defined as a R b if a > b
R ={(2,1),(3,1),(3,2),(4,1),(4,2),(4,3)}
Here (1,1) ,(2,2),(3,3) and (4,4,) are not exist in the relation R.
So R is not reflexive.
Example 2
Consider the above example 2 , A relation R in the set of integers is defined as
a R b if 2 divides a-b
R={(0,0),(0,2),(0,4)....,(1,1),(1,3),(1,5)......,(2,0),(2,2),(2,4),....,....(0,-2),
(0,-4)....}
Here , for all a ϵ Z, a – a = 0 is divisible by 2. So R is reflexive.
4. Symmetric Relation
A relation R on a set A is said to be symmetric relation if and only if
(a , b) ϵ R then (b , a) ϵ R, for all a , b ϵ A
Working Rule to check symmetric:
(i)
(ii)
Take two elements from the given set
Assume (a , b) ∈ R as per the given definition.Replace b by a and a
by b in the given definition of relation. Check whether (b , a) also
there in R.
If valid, then Symmetric. If not valid, not Symmetric
Example-1
Relation R defined on Set A = {1, 2, 3, 4} as (a , b) ∈ R if a > b
The set R = {(2 , 1), (3 , 1) , (3 , 2) , (4 , 1), (4 , 2), (4 , 3)}
In this relation R,
For the elements 1 , 2 A, (2, 1) ∈ R But (1, 2) ∉ R.
Hence R is not symmetric.
Example-2
A relation R in the set of integers Z is defined as a R b if 2 divides a-b.
R={(0,0),(0,2),(0,4)....,(1,1),(1,3),(1,5)......,(2,0),(2,2),(2,4),....,....(0,-2),(0,4)....}
For a ,b ∈ Z, let (a , b) ∈R, Here if 2 divides a - b
⇒ 2 divides b - a also
Hence, (b , a)∈ R
In particular, (0,2) ∈ R ⇒ (2,0) ∈ R
 Hence R is symmetric
5. Transitive Relation:
A relation R on A is said to be a transitive relation iff (a, b)∈ R and
(b, c) ∈R ⇒(a, c) ∈R for all a, b, c ∈ A.
i.e. aRb and bRc ⇒aRc for all a,b,c ∈A
WORKING RULE TO CHECK TRANSITIVE RELATION
If the relation set R is finite,
 Take three elements from the given set
 Assume (a , b) ∈ R and (b , c)∈ R as per the given definition
 Check if (a , c) ∈ R using the above results
If valid then Transitive. If not valid, not Transitive
If the relation set R is infinite
Take three elements from the given set. Assume (a , b) ∈ R and (b , c)∈ R as
per the given definition
Check if (a , c) ∈ R using the definition of relation R
If valid then Transitive. If not valid, not Transitive.
Important Note: Let a relation R on A is defined such that
(a, b) ∈R and (b, c) ∉ R
 In such cases we need not check (a , c)∈R for all a, b, c ∈ A
 In such cases it is assumed that transitive relation is obviously true
Example-1
Relation R defined on Set of real numbers as (a , b) ∈ R if a > b or a R b if a >b
Let (a , b) ∈ R ⇒ a > b
Let (b , c) ∈ R ⇒ b > c
⇒a>c
Hence (a , c)∈ R
 Hence R is transitive
Example-2
A relation R in the set of integers is defined as a R b if 2 divides a-b
R={(0,0),(0,2),(0,4)....,(1,1),(1,3),(1,5)......,(2,0),(2,2),(2,4),....,....(0,-2),(0,4)....}
 For a, b , c ∈ Z
 Let (a , b) ∈ R ⇒ 2 divides a – b ⇒ a – b = 2m
 Let (b , c) ∈ R ⇒ 2 divides b – c ⇒ b – c = 2n
 Now, a – c = (a – b) + (b – c) = 2m + 2n =2(m+n)
⇒ 2 divides a – c
 Hence (a , c)∈ R

 Hence R is transitive
6. Equivalence Relation
A relation R in a set A is said to be equivalence relation if it is
(i) Reflexive, ie if (a,a) ∈ R for all a ∈ A
(ii) Symmetric, ie if (a,b) ∈ R implies (b,a) ∈ R
(iii)Transitive, ie, if (a,b) ∈ R and (b,c) ∈R implies (a,c) ∈R for all a,b,c ∈ A.
Exmples.
T is the set of all triangles in a plane with a relation R on T is given by
R={(T1,T2):T1 is congruent to T2}.This relation R is an equivalence Relation
Reflexive: R is reflexive as every triangle is congruent to itself
Symmetric: R is symmetric as if triangle T1 is congruent to T2 then obviously
T2 is congruent to T1
Transitive: R is transitive because if T1 and T2 are congruent,T2 and T3 are
congruent, then T1 will definitely congruent to T3.
Hence, R is an equivalence relation