Reflexive Property

The reflexive property is a binary relation on a set, where every element is related to itself. 
For instance, a relation R on a set A is said to be reflexive if, for every element a ∈ A, the pair (a, a) is included in R. 
 

`A reflexive relation satisfies specific characteristics. Some properties of a reflexive relation are:

• An empty relation on a non-empty set is not reflexive, because it does not contain any pairs related to itself.

• A relation defined on an empty set is always reflexive, as there are no elements in the set.

• A universal relation on any set is always reflexive, as it includes all pairs (a, a) for every element in the set. 
   

In this section, let’s learn how to verify whether the relation is reflexive or not by following these steps:

Step 1: Identify the set
Consider a relation R defined on a set A

Step 2: Check self-pairs
Check each element a ∈ A to make sure the pair (a, a) is a part of the relation R.

Step 3: Conclusion
The relation R is reflexive if every element in A has its corresponding self-pair (a, a) in R.

The relation is not reflexive if even one self-pair is missing.

For example, for a set A = {1, 2, 3} and the relation R = {(1, 1), (2, 2), (3, 3), (1, 2), (2, 3)}
The pair (1, 1) is a part of R
The pair (2, 2) is a part of R
The pair (3, 3) is a part of R
As each element a ∈  A has the pair (a, a) in R, so R is reflexive on A. 
 

The reflexive property of congruence states any geometric figure is congruent to itself. In other words, a shape is always congruent to itself. It is represented by the symbol ≅. It is a fundamental concept in geometry and is used in geometric proofs.  
For example, two triangles △PQR and △SQR, where QR is the common side. If 
PQ = SQ
PR = SR
QR = QR (by the reflexive property of congruence)
So, △PQR ≅ △SQR 
 

The reflexive property of equality states any number is equal to itself. For example, x = x, 2 = 2, -8 = -8. 
The property is part of a relation R defined on the set of real numbers, where aRb if and only if a = b. This relation satisfies the three conditions necessary to be classified as an equivalence relation. 

Reflexive: For every real number a, aRa because a = a

Symmetric: If aRb, then b = a, so bRa

Transitive: If aRb and bRc, meaning a = b and b = c, then a = c, so aRc. 
 

A binary relation R on a set A is reflexive if every element in A is related to itself. For all elements a ∈ A, the pair (a, a) ∈ R or aRa. A relation is reflexive if each element of the set appears in a pair with itself within the relation. 
 

Many other relations are related to the reflexive property. Some related relations to the reflexive property are:

• Anti-Reflexive Relation  

• Co-reflexive Relation

• Left Quasi-Reflexive Relation

• Right Quasi-Reflexive Relation

Anti-Reflexive Relation: An anti-reflexive relation, also known as an irreflexive relation, is a type of binary relation in which no element in the set is related to itself. In other words, a relation R on a set A is anti-reflexive if, for every element a ∈ A, the pair (a, a) is not in the relation. This can be written as: ∀a ∈ A, (a, a) ∉ R.  

 
  
Co-reflexive Relation: A co-reflexive relation is also known as a covariant relation. It is a type of binary relation on a set where only identical elements are related, and both must be individually related to themselves. It can be represented as: if (a, b) ∈ R, then (a, a) ∈ R and (b, b) ∈ R. 

Left Quasi-Reflexive Relation:

A left quasi-reflexive relation is one where the binary relation on a set in which each element is related to itself from the left. For example, for every element “a” in the set, if it appears as the first element in any ordered pair (a, b) within the relation, then the pair (a, a) must be included in the relation. 

   
Right Quasi-Reflexive Relation: It is a type of binary relation on a set in which every element that appears as the second element in any ordered pair is also related to itself. This means that if element b appears as the right-hand side of any pair (a, b) in the relation, then (b, b) is in the relation. 
 

In the real world, the reflexive property is used in fields such as geometry, algebra, and identity verification, etc. Some applications of the reflexive property are:

• In computer security, we use the reflexive property to verify whether a user’s identity matches the stored data. This property is applied to login systems, biometric scanners, and digital signatures. 

• The reflexive property of congruence states whether the geometric figures are congruent or not. So we use it in 3D modeling software, CAD systems, and video game rendering. 

• In algebra, the reflexive property is used to solve equations and to simplify equations. We use it to prove other properties of equality.