Reflexive Relation

The reflexive pronoun in English grammar indicates the subject of the sentence. Similarly, in mathematics, reflexive relations are studied in the context of binary relations. The concept is applied to set theory to understand the relationship between two or more sets. Apart from reflexive relations, there are also symmetric and transitive relations. The ‘reflexive’ in reflexive relations stands for every set element that it reflects, that is, it indicates that every element in the set belongs to itself.

Thus reflexive relation can be defined as, 

‘For a set A there is a binary relationship R, and there exists a ϵ A. We have a Relation R such that (a, a) ϵ R  for each a ϵ A . (Here a represents every element of set A)’.

For example, let us consider a set A = {1,2,3,4,5,6}, and R is the relation defined as Z = {(1,1), (6,1), (4,4), (3,3), (2,2), (5,3), (5,5),(6,6)}. Since (1,1), (2,2), (3,3), (4,4), (5,5), and (6,6) belong to Z , we can say that it is a reflexive relation. Every element of A has its own image in set Z is known as a reflexive relation .

Determining the number of reflexive relations

For sets A and Z, we have a relation defined by R such that the n2 is the ordered pair of (a,z) in the set of relation R. For the n2 ordered pair, we have n2—n reflexive relations. Hence, a set with ‘n’ number of elements has a number of reflexive relations, which can be found by N = 2n(n-1).

Some important definitions

Irreflexive: Consider the elements of the set A, (a, a) ∉ R ∀ a, where R is the relation. In mathematics, the quasi reflexive relation is defined as if in a set A, there exist a ϵ A and z ϵ A such that (a,z) ϵ A and the relation is defined as R such that a R z implies a R a and z R z. The quasi reflexive relation is further divided into two categories:

1. Left quasi-reflexive: When in a set A, (a,z) A, the relation is defined on R such that (a, a) A ∀ (a,z) belongs to the set A.

2. Right quasi-reflexive: When in a set A, (a,z) A, the relation is defined on R such that (z,z) A (a,z) belongs to the set A.

Co-reflexive: It is a relation in which the relation R is defined such that for all elements of set A, (a,z) ϵ A and a R z, then a = z. The most common example of a co-reflexive relation is an equality relation. 

Solved examples

Consider a set A having all integers, and R is the relation defined by ‘a-b’ is divisible by 3, ∀ (a,b) A. Show that R is reflexive on A.

Solution. Let us consider an element ‘a’ in set A, such that a – a = 0. And 0 is divisible by 3, hence a R a for all elements in A, thus R is reflexive on set A.

In a set A, the relation R is defined by ‘if a ≥ b, ∀ A(a,b) , then a R b ‘. Show that R is reflexive on A.

Solution. Consider a set in which element ‘a’ belongs to the set A. According to the question, the relation is given by a ≥ b and we know that a = a. Hence a ≥ a which implies (a,a) R.
Thus the relation is reflexive.

Conclusion

Reflective relations in mathematics explain the relation in binary relations, to check the relationship of every element in the set with itself. The relation is reflexive if the image of every element in the set is present in the given set. To determine the number of reflexive relations the formula N = 2n(n-1) is applied. Where N is the number of reflexive relations a set has and n is the number of elements in a set. There are some important definitions like anti-reflexive, quasi-reflexive, and co-reflexive which are used in defining reflexive relations. We also discussed some examples to strengthen the concept of reflexive relations.