Reflexive Relation

What is a Reflexive Relation?

A reflexive relation is a binary relation that is reflexive, meaning that every element in the set is related to itself. For example, the relation “is taller than” is reflexive, because every element in the set of people is related to itself.

The Property of Reflexive Relations

Reflexive relations are those in which an object or element relates to itself. For example, a person’s height is a reflexive relation because a person’s height is always the same as the person’s height. Other examples of reflexive relations include a person’s weight, age, and gender.

Number of Reflexive Relationships

Reflexive relationships are those in which an object is both the subject and the object of the action. For example, when a person looks in a mirror, they are looking at themselves. Similarly, when a person writes a letter to themselves, they are both the subject and the object of the action.

Formula for Number of Reflexive Relations

The number of reflexive relations is the number of pairs of objects where each object is in a relation with the other object.

Reflexive Relation Characteristics

A reflexive relation is a relationship in which an object is both the subject and the object of the relationship. For example, in the relationship “I am writing a paper,” the object (the paper) is both the subject (the thing being written) and the object (the thing being written about). In general, reflexive relations are symmetrical; that is, the relationship is the same in both directions.

Reflexive relations have a few important characteristics. First, reflexive relations are always transitive; that is, if A is related to B and B is related to C, then A is also related to C. Second, reflexive relations are always symmetrical; that is, the relationship is the same in both directions. Finally, reflexive relations are always reflexive; that is, the relationship always holds between the object and itself.

Reflexive relations are important in mathematics, where they are used to define certain operations. For example, the operation of addition is defined as the reflexive relation that exists between two numbers, A and B, such that A + B = B + A. This means that the sum of two numbers is the same as the sum of the inverse of those numbers.

Reflexive Relation Examples

A reflexive relation is a binary relation (R) between two objects (x and y) such that xRy if and only if x = y.

Examples of reflexive relations include “being a parent of oneself,” “being a sibling of oneself,” and “being a student of oneself.”

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.

To know more about reflexive relations, log on to Infinity Learn and find out what the experts have to say. Understand the concepts from the easiest explanation given by the mentors and develop your conceptual foundation strongly.

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.