In mathematics, a relation on a set is called reflexive if every element in the set is related to itself. This means, for a set A, the relation R is reflexive if for every element a in A, the pair (a, a) is included in the relation. If even one element is not related to itself, then the relation is not reflexive.

Sets are collections of objects or elements. Relations and functions help us understand how elements from one set are connected to elements in the same or another set. A relation shows all possible pairings between elements of sets, while a function is a special type of relation where each input has only one output.

There are different types of relations such as empty relation, universal relation, identity relation, inverse relation, reflexive, symmetric, transitive, and equivalence relation—each describing a different way elements from sets can be connected.

What is Reflexive Relation?

In set theory, a binary relation on P is supposed to be a reflexive relation if each element of the set is related to itself. Let us suppose a mathematical example to understand the meaning of reflexive relations.

Specify a relation on the set of integers Z as ‘ is equal to’. Now, we remember that each integer is identical to itself such as 0 = 0, -2 = -2, 6 = 6, and so on. This implies every integer is related to itself. Hence, the relation ‘is equal to’ on the set of integers is a reflexive relation.

A binary relation R specified on a set A is supposed to be reflexive if, for each element a ∈ A, we have aRa, that is, (a, a) ∈ R. For example, if for p ∈ A, p is not related to itself then it is denoted by (p, p) ∉ R or ‘not pRp’.

Example of reflexive relation: Let X = {a, b, c, d, e} and R is a relation defined on X as R = {(a, a), (a, d), (b, b), (c, c), (d, d), (e, e), (d, e)}. Since, (a, a), (b, b), (c, c), (d, d), (e, e) ∈ R, therefore R is a reflexive relation as every element of X is related to itself in R.

If B = {1, 3}. Now, the reflexive relation will be R = {(1, 1), (3, 3), (1, 3), (3, 1)}. Hence, a relation is reflexive if:

(b, b) ∈ R ∀ b ∈ B

where b is the component, B is the set and R denotes the relation. The reflexive relation example is given in the table below. The statements consisting of these relations confer reflexivity.

We can define the number of reflexive relations on a set B. A relation R established on a set B with n elements has ordered pairs of the form of (a, b). Now, we identify that element ‘a’ can be taken in n ways and likewise, element ‘b’ can be taken in n ways. This implies we have \(n^{2}\) ordered pairs (a, b) in R.

For a reflexive relation, we require ordered pairs of the form (a, a) and (b, b). There are n-ordered pairs of the pattern (a, a), so there are \(n^{2} – n\) ordered pairs for a reflexive relation. Therefore, the total number of reflexive relations is\(2^{n\left(n-1\right)}.\)

Reflexive Relation Formula

The number of reflexive relations on a set with n elements is presented by \(N=2^{n\left(n-1\right)}\) where capital N is the number of reflexive relations and small n is the number of elements in the set.

Characteristics of Reflexive Relation

Reflexive relation is a relation of elements of a set A such that every element of the set is related to itself. As the name ‘ reflexive relations’ suggests, the image of every element of the set is its reflection.

1. Anti-Reflexive Relation: A relation R specified on a set P is said to be an anti-reflexive relation if none of the elements of P is related to itself. Mathematically, it is represented by the notation as (p, p) ∉ R for every p ∈ P. It is also understood as irreflexive relation.
2. Co-reflexive Relation: A relation R specified on a set Q is said to be a co-reflexive relation if (p, q) ∈ R ⇒ p = q for all p, q ∈ Q.
3. Quasi-reflexive Relation: A relation R marked on a set P is supposed to be a quasi-reflexive relation if (p, q) ∈ R ⇒ (p, p) ∈ R and (q, q) ∈ R for all p, q ∈ P.
4. Left Quasi-reflexive Relation: A relation R illustrated on a set A is stated to be a left quasi-reflexive relation if (a, b) ∈ R ⇒ (a, a) ∈ R for all a, b ∈ A.
5. Right Quasi-reflexive Relation: A relation R specified on a set A is supposed to be a right quasi-reflexive relationship if (a, b) ∈ R ⇒ (b, b) ∈ R for all a, b ∈ A.

Example of Reflexive Relations: Reflexive relation is a significant concept in set theory. For example, if there is a group of kids who do not possess siblings and the relation is determined as ‘is a sibling of’, then each child is its sibling, that is, each child is linked to itself. Hence, the relation is reflexive. Check out some of the reflexive relation example set:

• Is equal to(=) is a reflexive relation illustrated on a set P as each element of a set is equal to itself.
  •  pRp as p = p for all p ∈ P
• Greater than or equal to(≥) is a reflexive relation specified on a set P as all elements of a set are greater than or equivalent to itself.
  • pRp as p ≥ p for all p ∈ P
• Less than or equal to(≤) is a reflexive relation defined on a set P as every component of a set is less than or equivalent to itself.
  • pRp as p ≤ p for all p ∈ P
• Divides (/) is a reflexive relation described on a set P as every number divides itself. pRp as p/ p for all p ∈ P.

Properties of a Reflexive Relation 

A reflexive relation on a set means that every element is related to itself. If a set has elements like a, b, and c, then the reflexive relation must include (a, a), (b, b), and (c, c).

Now let’s understand some important properties:

1. An Empty Relation on a Non-Empty Set is Not Reflexive

• If a set has elements but the relation has no pairs (it's empty), then it's not reflexive.
• This is because for a relation to be reflexive, it must at least have (a, a) for all elements a in the set.
• So, if the set has items but none of the (a, a) pairs are in the relation, it cannot be reflexive.

Example:
Let A = {1, 2}
Empty relation: R = {}
Since (1,1) and (2,2) are missing, R is not reflexive.

2. A Relation on an Empty Set is Always Reflexive

• This might seem odd, but if the set itself is empty, there are no elements to check for reflexivity.
• So, by default, we say the relation is reflexive because there’s nothing to violate the condition.

Example:
A = ∅ (empty set)
R = ∅
Since there are no elements, all required reflexive pairs are “already there,” so it is reflexive.

3. A Universal Relation is Always Reflexive

• A universal relation includes all possible ordered pairs of a set (every combination of elements).
• That means it definitely includes all (a, a) pairs.
• So, it always satisfies the reflexive condition.

Example:
A = {1, 2}
Universal relation: R = {(1,1), (1,2), (2,1), (2,2)}
This relation has (1,1) and (2,2), so it is reflexive.

Applications of Reflexive Relation

Reflexive relations are widely used in various areas of mathematics, computer science, and real-life situations. Below are some important applications:

1. Mathematics and Set Theory
   Reflexive relations are used to define important concepts like equivalence relations and partial order relations. These are essential in organizing and comparing elements in a set.
2. Computer Science
   In database systems, reflexive relations are used when designing relationships where an entry might relate to itself. For example, an employee supervising themselves in a database table must follow a reflexive condition.
3. Graph Theory
   Reflexive relations appear in graphs where each node has a loop to itself. This is helpful in modeling self-loops in systems or processes.
4. Automata and State Machines
   Reflexive relations help define transitions where a machine can stay in the same state without moving to a different one.
5. Logic and Proofs
   Reflexive properties are used in mathematical proofs to show that every element is related to itself, such as in identity proofs or in reflexive closure of relations.
6. Decision-Making and Preference Systems
   In economics or operations research, reflexive relations model situations where a choice is as good as itself, helping build preference models.

Important Points on Reflexive Relations

A relation is a connection between sets of values. In math, the relation is amongst the x-values and y-values of ordered pairs. The set of all x-values is termed the domain, and the set of all y-values is named the range.

• A binary relation R specified on a set P is declared to be reflexive if, for every element p ∈ P, we have pRp, that is, (p, p) ∈ R.
• The number of reflexive relations on a set with the ‘n’ number of elements is presented by the formula \(2^{n\left(n-1\right)}\).
• A relation R defined on a set A is said to be an anti-reflexive relation if no element of A is related to itself.

Learn about Difference Between Relation and Function

Solved Examples of Reflexive Relation

Let us work on some solved examples of reflexive relations:

Example 1: What is the possible number of reflexive relations on a set of 5 elements?

Data: Number of elements in a set = n = 5

Formula: Total number of reflexive relations in a set = \(2^{n\left(n-1\right)}=2^{n^{2}-n}\).

Calculation: Total number of reflexive relations in a set =\(2^{n^2-n}=2^{5^2-5}=2^{25-5}=2^{20}\).

Example 2: Consider the following statements about reflexive and irreflexive relations:

• A: The set of all reflexive relations is closed under the operation of union.
• B: The set of all irreflexive relations is not closed under the operation of union.
• C: The set of all reflexive relations is not closed under set difference.
• D: The set difference of all reflexive relations is not irreflexive.

Question: Which of the above statements are false?

• Statement A: True.
  If two relations are reflexive, their union will also contain all diagonal pairs like (a, a). So, the union will be reflexive.
• Statement B: False.
  If you take the union of two irreflexive relations, it might include a diagonal pair (a, a), making it no longer irreflexive. So the set of irreflexive relations is not closed under union — this statement is true as written, so it is not false.
• Statement C: True.
  If you subtract one reflexive relation from another, you may remove some (a, a) pairs. So, the result may not be reflexive.
• Statement D: False.
  The difference of two reflexive relations may remove all (a, a) pairs, resulting in an irreflexive relation. So, the set difference can be irreflexive. This statement is false.

Statements B and D are false. So, the correct answer is: Both B and D.

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Example 3: How many reflexive relations are there on a set with 4 elements?

Data: Number of elements in a set = n = 4

Formula: Total number of reflexive relations in a set = \(2^{n\left(n-1\right)}=2^{n^{2}-n}\)

Calculation: Total number of reflexive relations in a set = \(2^{n^2-n}=2^{4^2-4}=2^{16-4}=2^{12}\).

We hope that the above article on Reflexive Relations is helpful for your understanding and exam preparations. Stay tuned to the Testbook App for more updates on related topics from Mathematics, and various such subjects. Also, reach out to the test series available to examine your knowledge regarding several exams.