Reflexive Relation

A reflexive relation has each element paired with itself. For instance, if A = {x, y}, then the relation must include (x, x) and (y, y). If any of the self-pairs is missing, then the relation cannot be reflexive. Reflexive relations are important in set theory, mathematics, and logic in concepts such as equality, similarity, and more.

Reflexive relations and identity relations both have a similar condition, that is the presence of self-pairs in the elements of their sets; however, there are a few key differences between them.

A relation can be proved as reflexive by following these steps:

1. List the elements of the base set A that form the basis for relation R.
   
    
2. Make sure that the relation R has a pair (a, a) for all a in the set A. 
   
    
3. The relation is reflexive only if every element has its own pair; if even one of the elements doesn't appear in a self-pair, the relation is not considered reflexive.

For example, for A = {4, 2, 5}, if R = {(4, 4), (2, 2), (5, 4), (4, 2), (5, 5)}, the relation is reflexive. 

Key properties of reflexive relations include:

• Every element relates to itself.

• Just because (a, b)(a, b)(a, b) is in R, it doesn't mean (b, a)(b, a)(b, a) has to be. Reflexivity does not guarantee symmetry. 

• A relation that is reflexive, symmetry, and transitive is called an equivalence relation.

• If a non-empty set has an empty relation, it is not reflexive.

• A relation defined on an empty set is always reflexive.

• A universal relation defined on any set is always reflexive.

To find how many reflexive relations can be formed on a set with n elements, follow these steps:

• A set A with n elements will have n² ordered pairs possible in a relation because each element can pair with all elements in the set.

• To be reflexive, a relation requires all self-pairs (a, a) for every a in set A. 

• Out of n², we know that n pairs are self-pairs; this leaves us with n² - n. The remaining pairs are non-diagonal, like (a, b), where a ≠ b. These remaining pairs are optional, giving us the option to include or not include them. So, the total number of reflexive relations on a set with n elements is 2^n2-n.

Let’s take an example for the same: A = {1, 2, 3}, so n = 3

Total possible pairs = 3² = 9

Required self-pairs are (1, 1), (2, 2), (3, 3)

Remaining pairs = 9 - 3 = 6

These remaining 6 pairs can either be included or not.

So, the number of reflexive relations = 2⁶ = 64

The scope of reflexive relations extends to various fields like computer science, law, and social systems. Some practical applications of reflexive relations are listed below:

• Identity verification in databases
  
  Reflexive relations ensure that each record in a database can be recognized as linked to itself. For example, a user ID is linked to the same user for authentication.

• Equality checks in software
  
  The equality relation is reflexive, as a = a. This property helps in object-oriented programming, where any object is equal to itself.

• Communication on social networks
  
  Social media apps allow users to send messages to their accounts, which shows a reflexive relation as the user is related to themselves.

• Legal records and citizenship status
  
  Citizen records in legal databases are linked to their identity, making it easy to organize and access their data. This process works on the principle of reflexivity.

• File ownership in computer operating systems
  
  Every user has ownership of their files in a computer system that they can access. Reflexive relation allows the owners access rights and holds accountability for data security.