All you need to know about Reflexive Relation

There are three types of relationships: reflexive, symmetric, and antisymmetric. The reflexive relations can be defined as (a, a) R for each and every “a” in the set X. The symmetric reflexive relation is defined as I, R for the mirror image of itself, where I would be the identity relation for A. The non-symmetric- symmetric factors that will impact its inverse are defined as I R, where I is the identity relation on A.

About reflexive relation

A reflexive relation is a relationship between components of a set A in which each component of the set is connected to one’s own. As the name implies, the image of each component of a set itself is the reflection of the component. In set theory, a reflexive relation is indeed an essential term. The connection “subgroup” of different set components, for instance, every component of the subset is a reflexive relation since every set is a compound entity of its subset.

The binary relation ‘R’ specified on a set A is considered as reflexive if aRa is taken, that is, (a, a) R for each and every component an A. This means that such a relation established on a set is reflexive such that each component of the set is associated with itself. R will not be considered a reflexive relation when there is a single member in the set which is not connected to its own. For example, if R is not reflexive if for b A and b is not connected to itself which is symbolized by (b, b) R  also as ‘not bRb’).

Any reflexive relation on the set A is sometimes denoted as IA = (a, a): an A, wherein IA R and R are both established on the set A. 

If we observe the following example- 

•  Let the set value of A = a, b, c, d, e and Reflexive value R = (a, a), (a, b), (b, b), (c, c), (d, d), (e, e), (c, e). Because (a, a), (b, b), (c, c), (d, d), (e, e) R, R is a reflexive relation because each member of A is connected with itself in R.

Common examples of Reflexive relations 

Reflexive meaning in set theory is explained by, a binary relation R over a set A is reflexive if every component of set A is connected or linked to itself. In the perspective of unions, this may be expressed as (a, a) R an A or as I R, wherein I is the identity relation on A.

The reflexive relation ‘Is similar to’ is described on a set A as every component of a set being identical to itself. For any a, aRa can be mentioned as a = a.

The correlation ‘greater or equal to’ is characterized reflexively on a set A of numbers just like every component of a set is approximately equal to itself. aRa as the a for any and all variables of a.

The reflexive meaning of the relationship ‘less than or equal to’ on a set A of numbers is that every component of a set is less than or equal to itself. aRa as including all set values of a. 

The relationship ‘divides’ is established reflexively on a set A of integers since each integer differentiates by itself.

The formula for determining the count of reflexive relations

N = 2n(n-1) gives the count of reflexive relations on a set with ‘n’ components, where N represents the number of reflexive relations, n is the total number of components in the set.

Amount of Reflexive relations

The amount of any component in the set and relations is an important attribute to determine the relation of the identity to the set value. On a set A, we may count the number of reflexive relations. A relation R formed on an n-element set A has sorted pairs of the type (a, b). We now understand that component ‘a’ may be adopted in n different ways, and conversely, component ‘b’ can be selected in n different ways. This means that R contains n2 sorted pairs (a, b). We require sorted pairs of the kind for a reflexive connection (a, a). There are n sorted pairings of the type (a, a), hence a reflexive relation has n2 – n ordered pairs. As a result, the maximum count of reflexive relationships is 2n (n-1).

Difference between reflexive relation and the identity relation

The conspicuous differences between Reflexive and Identity relation is stated by an example below- 

1. Let A represent any set. The relation R=(x,x):xA on A is thus known as the identity relation on A. As a result, in an identity relation, each component is solely connected to its own.

Consider the relation A=a,b,c and describe R and R’ as follows. R={(a,a),(b,b),(c,c)}

R’={(a,a),(b,b),(c,c),(a,c)}

Therefore R is an identity relation on A, but R’ is not since the element an is connected to a and c.

As observed above, in identity relations, each component of a set is associated with its own. 

2. A reflexive relation R on a set A is one in which every component of A is connected to itself. As a result, R is reflexive iff (x,x)R for every xA. If there is a component xA so that (x,x)R, a relation R of set A is not reflexive. Considering the expression A= (1,2,3). Then R is a reflexive relation on A, denoted as R=(1,1),(2,2),(3,3),(1,3),(2,1). R’=(1,1),(3,3),(2,1),(3,2) is not a reflexive relation on A, since (2,2)R2.

Any identity connection on a quasi set A is a reflexive relation, and it is not the other way around.   R is therefore a reflexive relation on A but it cannot be deemed as an identity relation.

Conclusion

Reflexive relations are a fundamental feature of set theory. A relationship that is homogenous R on a set X is reflexive if it refers to each and every element of X. The relation ‘is similar to the set of real numbers is an instance of a reflexive set and relation, because every real number is identical to itself.