Reflexive Relation

The relationship between any of the two elements of the set is known as a binary relationship. The binary relationship is determined to be equivalent when it is symmetric, transitive, and reflexive.  The binary relationship is the reflexive relationship when every element in the set S is linked with itself.  

In terms of maths, it is represented as  (a, a) ∈ R ∀ a ∈ S (or) I ⊆ R. In this sequence a is the element, R is the relation, and S is the set. It represents identity relation on the A. For instance, considering the set C = {7,9}. In this the reflexive relation is R = { (7,7) , (9,9), (7,9), (9,7) }. The set of real numbers is also the reflexive set, since each element which is a real number, tends to be equal to itself.

The Property of Reflexive Relations

As per the reflexive property, (a, a) ∈ R for each a ∈ S. Here a is the element, S is the set, and R is the relation.

In any given set, there are various reflexive relations that can be possible. For instance consider the set S. This set consists of an ordered pair (p, q). The p can be selected in n number of ways and similarly with q. This is why, this set of the ordered pairs, is made up of n square pairs. 

According to the concept of the reflexive relationship, (p, p) should be included in these ordered pairs. It is worth noting that there are a total of the n pairs, for such (p, p) pairs. As a result of this, the number of the ordered pairs would be n square -n pairs. Thus, the total number of the reflexive relationships in the set 2n(n−1).

The formula related to the number of reflexive relations in the given set is denoted by N = 2n(n−1). In this equation, N denotes the total number of reflexive relations, whereas n denotes the number of elements. Some of the characteristics associated with reflexive relations are Anti- reflexive, Quasi - reflexive, and Co - reflexive.

Reflexive Relation Table

Number of Reflexive Relations

In a given set there are a number of reflexive relations that are possible. Let us consider a set S. This set has an ordered pair (p, q). Now, p can be chosen in n number of ways and so can q. Therefore, this set of ordered pairs comprises \[n^{2}\] pairs. As per the concept of a reflexive relationship, (p, p) must be included in such ordered pairs. Also, there will be a total of n pairs of such (p, p) pairs. As a result, the number of ordered pairs will be \[ n^{2} - n\] pairs. Hence, the total number of reflexive relationships in set S is \[2^{n(n-1)}\].

Formula for Number of Reflexive Relations

The formula for the number of reflexive relations in a given set is written as 

N = \[2^{n(n-1)}\]

Here, N is the total number of reflexive relations, and n is the number of elements.

Reflexive Relation Characteristics

Some of the characteristics of a reflexive relation are listed below: 

• Anti - Reflexive: If the elements of the set do not relate to themselves, they are said to be irreflexive or anti-reflexive. 

• Quasi-Reflexive: If each element is related to a specific component, which is also related to itself, then that relationship is called quasi-reflexive. If a set A is quasi-reflexive, this can be mathematically represented as ∀ a, b ∈ A: a ~ b ⇒ (a ~ a ∧ b ~ b).

• Co-Reflexive: The relationship ~ (similar to) is co-reflexive for all elements a and b in set A if a ~ b also implies that a = b.

It is impossible for a reflexive relationship on a non-empty set A to be anti-reflective, asymmetric, or anti-transitive.

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Reflexive Relation Examples

Example 1: A relation R on set A (set of integers) is defined by “x R y if 5x + 9x is divisible by 7x” for all x, y ∈ A. Check if R is a reflexive relation on A.

Solution: 

Consider x ∈ A.

Now, 5x + 9x = 14x, which is divisible by 7x.

Therefore, x R y holds for all the elements in set A.

Hence, R is a reflexive relationship.

Example 2: A relation R is defined on the set of all real numbers N by ‘a R b’ if |a-a| ≤ b, for a, b ∈ N. Show that the R is not a reflexive relation.

Solution: 

N is a set of all real numbers. So, b =-2 ∈ N is possible. 

Now |a – a| = 0. Zero is not equal to nor is it less than -2 (=b).

So, |a-a| ≤ b is false.

Therefore, the relation R is not reflexive.

Example 3: A relation R on the set S by “x R y if x – y is divisible by 5” for x, y ∈ A. Confirm that R is a reflexive relation on set A.

Solution: 

Consider, x ∈ S.

Then x – x= 0. Zero is divisible by 5.

Since x R x holds for all the elements in set S, R is a reflexive relation.

Example 4: Consider the set A in which a relation R is defined by ‘m R n if and only if m + 3n is divisible by 4, for x, y ∈ A. Show that R is a reflexive relation on set W.

Solution: 

Consider m ∈ W.

Then, m+3m=4m. 4m is divisible by 4.

Since x R x holds for all the elements in set W, R is a reflexive relation.