Types of relation

Types of Relations

There are ten types of relations:

Inverse relation

Let R be any relation between two sets, A and B. The inverse of R, represented by R^-1, is the relationship between B and A consisting of the pairs whose reversed values belong to R; thus, 

R^-1= {(b, a) : (a, b)∈R}
 

Example: 

Let A={4, 6, 10} and B={8, 10, 12} and define a binary relation R from A to B as follows:

For all (x, y) ∈A X B, (x, y) ∈R ⇒x/y. Write each R and R^-1 as a set of ordered pairs.
 

Solution: 

First, we will find the cartesian product of A x B

A x B={ (4,8), (4, 10), (4, 12), (6, 8), (6, 10), (6, 12), (10, 8), (10, 10), (10, 12) }

Now we have to find the relation, which means we have to find the pairs from the cartesian product which satisfies the condition, i.e., x/y(x is divisible by y )

R= { (4, 8), (4, 12), (6, 12), (10, 10) }

Now we have to find the R^-1 means if (y is divisible by x)

R^-1 = { (8, 4), (12, 4), (12, 6), (10, 10) }

 

Identity relation

It is said that a relation R in a set A is an identity relation denoted as I_A if every element in Set A can be related to itself only using R⊆A x A.

Here, I_A = { (x, x): x ∈A }
 

Example:

Let A = {1, 2, 3} , then find I_A. 
 

Solution:

First find the relation A x A,

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

Therefore, I_A = { (1, 1), (2, 2), (3, 3) }

 

Universal Relation

The relation between every element of a set A and every other element of a set A is called a universal relation or R = A x A.

Example:  Given a set A ={1, 2, 3} find the universal Relation.
 

Solution: R= A x A ={ (1, 1), (1, 2), (1, 3), (2, 1), (2, 2), (2, 3), (3, 1), (3, 2), (3, 3) } here is the universal relation R.

 

Void or Empty relation

It is opposed to universal relations. Void relations can be regarded as R is a null set, i.e., if R = ∅
 

Example: 

The relation R on the set A={1, 2, 3} by R= {x, y}: x*y >10}.
 

Solution:

A x A = { (1, 1), (1, 2), (1, 3), (2, 1), (2, 2), (2, 3), (3, 1), (3, 2), (3, 3) }

We will find the pairs that satisfy the given condition, i.e., x*y is greater than 10.

There are no such pairs that satisfy the condition hence,

R = {∅} or { }
 

Reflexive relation

There is only one set in this. When a binary relationship R on a set A is reflexive if aRa means the relation of "a" must be with itself, and "a" is the element of the set A for each "a", that is, if (a, a) ∈ R ∀ a ∈ A.

There is only one set in this. When a binary relationship R on a set A is reflexive, it means the relation of "a" must be with itself, and a is the element of the set A for each "a", that is, if (a, a)  ∈ R ∀ a ∈ A.
 

Example: 

Let A = {1, 2, 3} and R is a relation defined on A as R = { (1, 1), (1, 2), (2, 2), (3, 3), (3, 1) } find if it is reflexive or not.
 

Solution: 

In the diagram, we can see arrows pointing to the number that is connected to the relation R given in this question, i.e., 1->1, 2->2, 3->3, 1->2, 3->1.

We have ordered pairs that have relationships to themselves that are (1, 1), (2, 2), (3, 3); therefore, R is a reflexive relation since every element of A is related to itself in R.

 

Irreflexive relation

There is only one set in this. When a binary relationship R on a set A is reflexive if aRa means the relation of "a" must be with itself, and "a" is the element of the set A for each "a", that is, if (a, a) ∉ R ∀ a ∈ A.
 

Example:

Let A = {1, 2, 3} and R is a relation defined on A as R = { (1, 2), (3, 1) } find if it is reflexive or not.
 

Solution: 

The diagram shows arrows pointing to the number connected to the relation R given in this question, i.e., 1->2, 3->1.

Here ordered pairs that have relations are  (1, 2), (3, 1) ∈ R, but (1, 1), (2, 2), (3, 3) are missing; therefore, R is an irreflexive relation as every element of A is related to itself in R.

 

Symmetric relation

A relation is supposed to be a symmetric one, in which all the ordered pair has given set plus the reverse ordered pair are present in the relation.

A relation R is symmetric on a set A if whenever a is related to b (aRb), then b is related to a (bRa) here a and b are the elements of the set A.

Therefore,

if (a, b)∈R then (b, a) ∈ R ∀ a, b ∈A

It means that if we have any ordered pair in R, which also has the reverse of its ordered pair, it is known as a symmetric relation.
 

Example:

Identify that the given relation R={(1, 2),(3, 2),(2, 1),(2, 3)} on set A={1,2,3} is symmetric or not.
 

Solution:

In the diagram, we can see arrows pointing to the number connected to the relation R given in this question, i.e., 1->2, 2->1, 3->2, 2->3.

For a Relation to be symmetric, the condition (a, b)∈R then (b, a) ∈ R ∀ a, b ∈A must be satisfied.

Here we can see that in relation R, we have (3, 2), (1, 2) ∈ R and (2, 3), (2, 1)∈R ∀ 1, 2, 3 ∈A. hence R is symmetric.

 

Antisymmetric relation

An antisymmetric relation on a set A occurs when a is related to b (aRb), b is related to a (bRa), then a = b. It means if there is no pair of distinct elements of A related to each other by R.

Therefore, if (a, b)∈R then (b, a)∈R then a = b ∀ a, b ∈A

also,

For all a, b ∈ A, If (a,b) ∈ R and a ≠ b, then (b,a) ∈ R must not be true.
 

Example: 

Determine if the following relation R is antisymmetric defined on set A = {1, 2, 3} R={(3,3),(1,1),(2,1),(1,3)}.
 

Solution:

The diagram shows arrows pointing to the number connected to the relation R given in this question, i.e., 1->1, 3->3, 2->1, 1->3.

So we can say that there is no pair of distinct elements of A that are related to each other by R. For every a, b ∈ A, (a, b) ∈ R and (b, a) ∈ R only when a = b. Therefore, R is an antisymmetric relation.

 

Transitive relation

A transitive relation is an association between elements on a given set. Suppose the first component is correlated to the second component, and the second component is correlated to the third component. In that case, the first element must be correlated to the third element.

Any binary relation R on a set A is transitive as long as aRb (a is related to b) is followed by aRc (a is related to c), where a, b, and c are the elements of the set A. 

Therefore, if (a, b)∈R, (b, c)∈R then 

         (a, c)∈R ∀ a, b, c ∈A 
 

Example:

Determine if a set A = {1, 2, 3} and R be a relation defined on set A and R ={(1, 2), (2, 3), (1, 3)} is transitive or not.
 

Solution:

Here a=1,b=2,c=3

The diagram shows arrows pointing to the number connected to the relation R given in this question, i.e., 1->2, 2->3, 1->3.

For the relation to be transitive, the condition (1, 2) ∈ R, (2, 3) ∈ R then (1, 3) ∈ R ∀ 1, 2, 3 ∈ A is being followed; hence the Relation R is transitive.

 

Equivalence relation

A binary equivalence relation is defined on X. It is reflexive, transitive, and symmetric. An equivalence relation on a set A is a subset of A×A, meaning an ordered pair of elements of A certain satisfying properties. We say "a is related to b" by writing "aRb" when (a,b) is an element of R. A relationship of equivalence is generally marked with the symbol '*' or '='.
 

Example:

Determine if the set set A={1, 2, 3} and the relation R = { (1, 2), (2, 1), (1, 1), (2, 2), (3, 3) } is equivalence relation or not.
 

Solution:

In the diagram, we can see arrows pointing to the number that is connected to the relation R given in this question, i.e., 1->2, 2->1, 1->1, 2->2, 3->3.

If the relation follows the equivalence relation or not, then it must follow the following three properties:

1. Reflexive: (1, 1), (2, 2), (3, 3) ∈ R, therefore R is a reflexive relation.
2. Symmetric: (1, 2)  R and (2, 1) ∈ R ∀ 1, 2 ∈ A. hence R is symmetric. 
3. Transitive: (1, 2) ∈ R, (2, 1) ∈ R then (1, 1) ∈ R ∀ 1, 2, 3 ∈ A hence R is transitive.
    

Due to the existence of all three properties, the given relation is an equivalence relation.

FAQs

1. Relations may exist between?
   There may exist a relation between objects from the same set or several sets.
    
2. How are relations and functions different?
   When a relation is defined, it represents the relationship between the input and output of two sets. In contrast, when a function is defined, there can only be one output for each input in a function. It represents an output for each input.
    
3. What is the number of relations in a set?
   If we have two Set A and Set B, the number of different relations from a Set A with “a” elements to a set with “b” elements is 2^ab.

Key Takeaways

In this blog, we have seen the Types of Relations present in discrete mathematics like Inverse, Identity, Universal and Void relation.

We hope that this blog has helped you enhance your knowledge about Types of Relations and if you would like to learn more, check out our articles on the link.

Recommended Readings:

• Has-A Relationship in Java

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