In Maths, a binary relation R across a set X is reflexive if each element of set X is related or linked to itself. In terms of relations, this can be defined as (a, a) ∈ R ∀ a ∈ X or as I ⊆ R where I is the identity relation on A. Thus, it has a reflexive property and is said to hold reflexivity. Symmetry, transitivity and reflexivity are the three properties representing equivalence relations.
For example, when every real number is equal to itself, the relation “is equal to” is used on the set of real numbers. A reflexive relation is said to have the reflexive property or is meant to possess reflexivity.
Reflexive Relation Definition
In relation and functions, a reflexive relation is the one in which every element maps to itself. For example, consider a set A = {1, 2,}. Now, the reflexive relation will be R = {(1, 1), (2, 2), (1, 2), (2, 1)}. Hence, a relation is reflexive if:
| (a, a) ∈ R ∀ a ∈ A |
Where a is the element, A is the set and R is the relation.
The examples of reflexive relations are given in the table. The statements consisting of these relations show reflexivity.
| Statement | Symbol |
| “is equal to” (equality) | = |
| “is a subset of” (set inclusion) | ⊆ |
| “divides” (divisibility) | ÷ or / |
| “is greater than or equal to” | ≥ |
| “is less than or equal to” | ≤ |
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Reflexive Relation Characteristics
- Anti-reflexive: If the elements of a set do not relate to itself, then it is irreflexive or anti-reflexive.
- Quasi-reflexive: If each element that is related to some element is also related to itself, such that relation ~ on a set A is stated formally: ∀ a, b ∈ A: a ~ b ⇒ (a ~ a ∧ b ~ b).
- Co-reflexive: A relation ~ (similar to) is co-reflexive for all a and y in set A holds that if a ~ b then a = b. The combination of co-reflexive and transitive relation is always transitive.
- A reflexive relation on a non-empty set A can neither be irreflexive, nor asymmetric, nor anti-transitive.
Reflexive Relation Formula
Number of reflexive relations on a set with ‘n’ number of elements is given by;
| N = 2n(n-1) |
Suppose, a relation has ordered pairs (a,b). Here the element ‘a’ can be chosen in ‘n’ ways and same for element ‘b’. So, the set of ordered pairs comprises n2 pairs.
As per the definition of reflexive relation, (a, a) must be included in these ordered pairs. Also, there will be a total of n pairs of (a, a). Hence, a number of ordered pairs here will be n2-n pairs. Therefore, the total number of reflexive relations here is 2n(n-1).
Reflexive Relation Examples
Q.1: A relation R is on set A (set of all integers) is defined by “x R y if and only if 2x + 3y is divisible by 5”, for all x, y ∈ A. Check if R is a reflexive relation on A.
Solution: Let us consider x ∈ A.
Now 2x + 3x = 5x, which is divisible by 5.
Therefore, xRx holds for all ‘x’ in A
Hence, R is reflexive.
Q.2: A relation R is defined on the set of all real numbers N by ‘a R b’ if and only if |a-b| ≤ b, for a, b ∈ N. Show that the R is not reflexive relation.
Solution: The relation is not reflexive if a = -2 ∈ R
But |a – a| = 0 which is not less than -2(= a).
Therefore, the relation R is not reflexive.
Q.3: A relation R on the set A by “x R y if x – y is divisible by 5” for x, y ∈ A. Check if R is a reflexive relation on set A.
Solution: Let us consider, x ∈ A.
Then x – x is divisible by 5.
Since x R x holds for all x in A
Therefore, R is reflexive.
Q.4: Consider the set A in which a relation R is defined by ‘x R y if and only if x + 3y is divisible by 4, for x, y ∈ A. Show that R is a reflexive relation on set A.
Solution: Let us consider x ∈ A.
So, x + 3x = 4x, is divisible by 4.
Since x R x holds for all x in A.
Therefore, R is reflexive.
A relation R is defined on N. What is the reflexive relation.
A relation R is defined on N is said to be reflexive if, for every element a ∈ N, we have aRa, i.e., (a, a) ∈ R.