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. 

Here is an example of a reflexive relation.

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

1. An empty relation on a non-empty set is not reflexive, because it does not contain any pairs related to itself.
    
2. A relation defined on an empty set is always reflexive, as there are no elements in the set.
    
3. 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:

1.  Identify the set.
   Consider a relation R defined on a set A.
    
2.  Check self-pairs:
   Check each element a ∈ A to make sure the pair (a, a) is a part of the relation R.
    
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 a single 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 a R b 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. 

The reflexive property of relations are mentioned below:

1. A binary relation R on a set A is reflexive if every element in A is related to itself.
    
2. For all elements a ∈ A, the pair (a, a) ∈ R or aRa.
    
3. 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

1. 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.  
   
    
2. 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. 
   
    
3. 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. 
   
    
4. 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. 

To make reflexive property easy for younger students, here are a few tips and tricks:

1. Assume reflexive property as a mirror, the object and its reflection are the same.
    
2. When solving congruency, remember a line or an angle is always equal to itself.
    
3. Use real life objects to check reflexive property. For example, is 2 plants are equal to 2 plants. Yes.
    
4. Understand the clear difference between, reflexive, symmetric and transitive property.
   Reflexive: a = a
   Symmetry: a = b, then b = a.
   Transitive: If a = b and b = c, then a = c.
    
5. Remember, a relation defined on an empty set is always reflexive.

Tip for Parents: 
 

• Use the analogy of mirror to explain the definition of reflexive property.
   
• Take real life objects as an example, like toys, coins, pennies, food, etc.
   
• Encourage your child to notice reflexive property in basic things, like $1 = $1, etc.

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 log in 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. 

• In architecture, if two spaces are of equal measurement or share the same wall, the spaces are the same by reflexive property. This helps engineerings and architecture to calculate materials or area only by finding for one designated space. 

• In banking, law, and even elections, identification of an individual is also an example of reflexive property. It is made to ensure that the person in the documents and the one presenting it are the same to prevent identity theft, or any scams and frauds.