A reflexive relation is a type of binary relation on a set where every element in the set is related to itself. In other words, for any element "a" in the set, the pair (a, a) is a part of the relation. Formally, a relation R on a set A is reflexive if, for all elements "a" in A, (a, a) is in R.
Examples:
- The "is equal to" relation (denoted by "=") is a reflexive relation on the set of real numbers, because every real number is equal to itself.
- The "is a parent of" relation on the set of people is reflexive because every person is their own parent (in a biological sense).
(a, a) ∈ R ∀ a ∈ A, i.e. a R a for all a ∈ A, where R is a subset of (A x A), i.e. the cartesian product of set A with itself.
Reflexive Relation Meaning
This means if element “a” is present in set A, then a relation “a” to “a” (a R a) should be present in the relation R. If any such a R a is not present in R, then R is not a reflexive relation.
A reflexive relation is denoted as:
R = {(a, a): a ∈ A}
Example: Consider set A = {a, b} and R = {(a, a), (b, b)}. Here, R is a reflexive relation, as for both a and b, a R a and b R b are present in the set.
Examples of Reflexive Relations
A reflexive relation is a type of binary relation on a set where every element in the set is related to itself. Here are some examples of reflexive relations:
1) Equality: The relation of equality on any set is reflexive. For any element a in the set, (a, a) is in the relation.
For example, on the set of real numbers, if a = b, then (a, b) is in the relation.
2) "Is a multiple of" Relation: On the set of integers, the relation "is a multiple of" is reflexive because every integer is a multiple of itself.
For any integer a, (a, a) is in the relation.
3) "Is a subset of" Relation: On the set of all sets, the relation "is a subset of" is reflexive because every set is a subset of itself.
For any set A, (A, A) is in the relation.
4) "Is congruent modulo n" Relation: On the set of integers, the relation "is congruent modulo n" is reflexive because every integer is congruent to itself modulo any integer n.
For any integer a, (a, a) is in the relation.
Number of Reflexive Relations
The number of reflexive relations on an n-element set is given by the formula:
2n(n-1)
Reflexive Relation Formula
To represent the number of reflexive relations on a set with n elements mathematically, you can use the following formula:
Number of Reflexive Relations = 2(n(n-1))
Properties of a Reflexive Relation
- An empty relation on a non-empty relation set is never reflexive.
- A relation defined on an empty set is always reflexive.
- A universal relation defined on any set is always reflexive.
How to verify a Reflexive Relation?
To verify whether any relation is reflexive or not, we can use the following steps:
- Check for the existence of every a R a tuple in the relation for all a present in the set.
- If every tuple exists, only then the relation is reflexive. Otherwise, not reflexive.
Follow the illustration below for a better understanding:
Example: Consider set A = {a, b} and a relation R = {{a, a}, {a, b}}.
For the element a in A:
⇒ The pair {a, a} is present in R.
⇒ Hence aRa is satisfied.
For the element b in A:⇒ The pair {b, b} is not present int R.
⇒ Hence bRb is not satisfied.
As the condition for ‘b’ is not satisfied, the relation is not reflexive.
Reflexive Relation: Related Relation
Some related relations to reflexive relation are:
Anti-Reflexive Relation
An antireflexive relation (also known as an irreflexive relation) is a binary relation on a set where no element is related to itself. In other words, for all elements a in the set, the pair (a, a) is not in the relation.
To express this concept more formally, a relation R on a set A is antireflexive if and only if for all elements a in A, the following statement is true:
∀ a ∈ A, (a, a) ∉ R
Co-Reflexive Relation
A co-reflexive relation (or covariant relation) is a binary relation on a set where all elements that are related to each other must be related to themselves. In other words, if (a, b) is in the relation, then both (a, a) and (b, b) must also be in the relation.
Formally, a relation R on a set A is co-reflexive if and only if for all elements a and b in A, the following statement is true:
If (a, b) ∈ R, then both (a, a) and (b, b) must be in R.
Left Quasi-Reflexive Relation
A left quasi-reflexive relation is a binary relation on a set where every element in the set is related to itself from the left side, but not necessarily from the right side. In other words, for all elements a in the set, there is a requirement that (a, a) is in the relation, but it is not necessary for (a, a) to be in the relation for every element from the right side of the pair.
Right Quasi-Reflexive Relation
A right quasi-reflexive relation is a binary relation on a set where every element in the set is related to itself from the right side, but not necessarily from the left side. In other words, for all elements a in the set, there is a requirement that (a, a) is in the relation, but it is not necessary for (a, a) to be in the relation for every element from the left side of the pair.
Related Articles:
Solved Problems on Reflexive Relation
Problem 1: Consider a set A = 1, 2, 3, and let R be a relation on A defined by R = (1, 1), (2, 2), (3, 3), (1, 2), (2, 1). Determine whether the relation R is reflexive.
Solution:
A relation is reflexive if every element in the set is related to itself. In this case, we need to check whether (a, a) is in R for every a in A. Let's check:
- For a = 1, (1, 1) is in R.
- For a = 2, (2, 2) is in R.
- For a = 3, (3, 3) is in R.
Since every element in A is related to itself, the relation R is reflexive.
Problem 2: Let B be the set of all people, and define a relation S on B by S = (x, y) mid x is a sibling of y. Determine whether the relation S is reflexive.
Solution:
For a relation to be reflexive, every element in the set must be related to itself. In the context of siblings, this means every person is a sibling to themselves, which is not true. Therefore, the relation S is not reflexive.
Problem 3: Consider the set C = a, b, c, and define a relation T on C by T = (a, a), (b, b), (c, c), (a, b), (b, a), (b, c), (c, b). Determine whether the relation T is reflexive.
Solution:
To check if T is reflexive, we need to ensure that (a, a), (b, b), and (c, c) are all in T. Let's check:
- (a, a) is in T.
- (b, b) is in T.
- (c, c) is in T.
Since every element in C is related to itself, the relation T is reflexive.
Reflexive Relations Practice Questions
Question 1: Consider the set {1, 2, 3}. Find all possible reflexive relations on this set.
Question 2: On the set of real numbers, define a new relation T as follows: (x, y) is in relation T if and only if x = y^2. Is relation T reflexive?
Question 3: On the set of integers, define a relation U as follows: (m, n) is in relation U if and only if m divides n. Is relation U reflexive?
Question 4: Determine whether the following relations are reflexive or not:
- R = {(x, x) | x is a prime number}
- S = {(a, a) | a is an even integer}