 # Reflexive Relation

A relationship between two elements of a set is called a binary relationship. A binary relationship is said to be in equivalence when it is reflexive, symmetric, and transitive. A binary relationship is a reflexive relationship if every element in a set S is linked to itself. In mathematical terms, it can be represented as (a, a) ∈ R ∀ a ∈ S (or) I ⊆ R. Here, a is an element, S is the set and R is the relation. I is the identity relation on A. For example, let us consider a set C = {7,9}. Here the reflexive relation will be R = {(7,7), (9,9), (7,9), (9,7)}. A set of real numbers is also a reflexive set, because each element i.e. each real number “is equal to” itself.

### Reflexive Property of Relations

According to the reflexive property, (a, a) ∈ R, for every a ∈ S, where a is an element, R is a relation and S is a set. For instance, let us assume that all positive integers are included in the set X. Now, for all pairs of positive integers in set X, ((p,q),(p,q))∈ R. Then, we can say that (p,q) = (p,q) for all positive integers.

This proves the reflexive property of equivalence.

## Reflexive Relation Table

 Statement Symbol “is equal to” (equality) = “is a subset of” (set membership) ⊆ “divides” (divisibility) ÷ or / “is greater than or equal to” ≥ “is less than or equal to” ≤

### 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 of n2 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 n2-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 that 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.

### 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.