Reflexive Relation: Overview, Questions, Preparation

Any relation can be termed as reflexive relation on a particular set when every element of that set is related to itself. 

In mathematics, R is the binary relation across the set. X can be termed as reflexive when every set X element is linked or related to itself. Reflexivity, transitivity, and symmetry are three distinct properties that represent equivalent relations.

A reflexive relation in relation and function is where each element maps with itself. For instance, if set A = {1,2} thus, the reflexive relation R = {(1,1), (2,2) , (1,2) , (2,1)}. Therefore, the relation is reflexive when :

(a, a) ∈ R ∀ a ∈ A

Here a is an element, R is the relation, and A is the set. 

The reflexive relations are mentioned below, where the statements depict reflexivity.

Characteristics of Reflexive Relation 

• Anti-reflexive: A relation is irreflexive or anti-reflexive if and only if the set's elements do not relate to itself.
• Quasi-reflexive: If each element that is related to some element is also related to itself, such that relation ~ on a set R is stated formally: ∀ a, b ∈R: a ~ b ⇒ (a ~ a ∧ b ~ b).
• Co-reflexive: A relation ~  is co-reflexive for ∀a and y in set A holds that if a ~ b, then a = b. 
• A reflexive relation on the non-empty set B can neither be irreflexive, nor asymmetric, nor anti-transitive.

Reflection relation is an important topic in relation and functions; students must study the topic thoroughly as it will help higher education. This chapter is covered in class 11 and holds the weightage of 6 marks.

1. Let A = {0, 1, 2, 3} and Let a relation R on A as follows: R = {(0, 0), (0, 1), (0, 3), (1, 0), (1, 1), (2, 2), (3, 0), (3, 3)}.
Show whether R is reflexive, Symmetric, or Transitive?

Solution.

R is reflexive and symmetric relation but not the transitive relation since for (1, 0) ∈ R and
(0,3)∈R, whereas the pair (1,3) ∉R.

2. Let set B in a relation P be defined by ‘xPy iff x + 3y, which is divisible by 4, for x, y ∈ B. Prove that P is a reflexive relation on the set B.

Solution.
Let x ∈ B. Now x + 3x = 4x, which is divisible by 4. Therefore xRx holds for all x in B, i.e., P is reflexive.

3. A relation P is defined on all real numbers R by ‘aPc’ iff |a – b| ≤ b, for a, b ∈ R. Prove that the P is not a reflexive relation.
Solution. 
The P is not reflexive as a = -2 ∈ R but |a – a| = 0 which is not less than -2(= a).

4. What is the formula of reflexive relations?

Solution. N = 2n(n-1), where n is the total number of elements in a set.

Q: What is the meaning of reflexive relations?

Q: When can a relation be termed as reflective?

Q: How to prove reflexive relation?

Q: Who invented the sets?

Q: What is the formula of reflexive relations?