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Reflexive Relation is a relation between elements of a Set A in which each element is related to itself. In a reflexive relation, each element of the set has its own reflection.
- Reflexive Relation is an integral concept in the Set Theory.
- Since every set is a subset of itself, thus, the relation “is a subset of” on a group of sets denotes a Reflexive Relation.
- There are different types of relations in discrete mathematics such as Reflexive, Symmetric, Transitive, etc.
A relation is considered to be a Reflexive Relation if
| (a, a) ∈ R ∀ a ∈ A |
Where a refers to the Element, A denotes the Set and R is the Relation.
Read More: NCERT Solutions For Class 12 Mathematics Relations and Functions
| Table of Content |
Key Terms: Reflexive Relation, Reflexivity, Relations, Set Theory, Sets, Symmetry, Transitivity, Reflexive Relation Formula, Ordered Pair
What is Reflexive Relation?
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Reflexive Relation is defined as a relation in which every element maps to itself.
- It is said to have the reflexive property or possess reflexivity.
- It is one of the three properties defining Equivalence Relations along with Symmetry and Transitivity.
Example: Consider a set A = {1, 2}. The reflexive relation will be R = {(1, 1), (2, 2), (1, 2), (2, 1)}.
Thus, a relation is a reflexive relation if
| (a, a) ∈ R ∀ a ∈ A |
Where
- a denotes the Element.
- A denotes the Set.
- R denotes the Relation.
Relations and Functions Detailed Video Explanation
Reflexive Relation Definition
In Set Theory, a binary relation R defined on Set A is said to be a reflexive relation if every element of the set is related to itself.
- If there is even a single element of the set that is not related to itself, then R is not a reflexive relation.
- For instance, if for b ∈ A and b are not related to itself (Denoted by (b, b) ∉ R or 'not bRb'), then R is not a reflexive relation.
- Reflexive Relation on a Set A is also expressed as IA = {(a, a): a ∈ A}, where IA ⊆ R and R is a relation defined on Set A.
Example: Consider Set A = {a, b, c, d, e} and R is a relation defined on A. Thus,
R = {(a, a), (a, b), (b, b), (c, c), (d, d), (e, e), (c, e)}.
As (a, a), (b, b), (c, c), (d, d), (e, e) ∈ R, thus, R is a reflexive relation as every element of A is related to itself in R.
Read More:
| Relevant Concepts | ||
|---|---|---|
| Relations and Functions | Real Valued Functions | What is a Function? |
| Types of Functions | Bijective Function | Difference between Relation and Function |
Examples of Reflexive Relations
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Given below are the examples of Reflexive Relations:
| Statement | Symbol | Explanation |
|---|---|---|
| “is equal to” (Equality) | = | Every element of a set is equal to itself. |
| “is a subset of” (Set Inclusion) | ⊆ | Every set is a subset of itself. |
| “divides” (Divisibility) | ÷ or / | Every number divides itself. |
| “is greater than or equal to” | ≥ | Every element of a set is greater than or equal to itself. |
| “is less than or equal to” | ≤ | Every element of a set is less than or equal to itself. |
Number of Reflexive Relations
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The number of Reflexive Relations on a Set A can also be determined.
- Relation R defined on a Set A with n elements has ordered pairs of the form of (a, b).
- It is known that element 'a' and element ‘b’ can be chosen in n ways.
- It means that there are n2 ordered pairs (a, b) in R.
- In a Reflexive Relation, ordered pairs of the form (a, a) are required.
- There are n ordered pairs of the form (a, a), thus, it can be said that there are n2 - n ordered pairs for a reflexive relation.
- Thus, the total number of reflexive relations will be 2n(n-1).
Read More: Relations & Functions Class 12 Important Questions
Reflexive Relations Formula
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Reflexive Relations Formula is used to find the Number of Reflexive Relations on a Set A with ‘n’ number of elements. The theory behind the formula has already been described in detail above.
Reflexive Relations Formula is given as:
| N = 2n(n-1) |
Where
- N: Number of Reflexive Relations
- n: Number of Elements in Set
Solved ExampleExample: What will be the number of Reflexive Relations from Set A to A, defined as A = a, b, c? Solution: Given that, A = a, b, c Thus, the number of elements in Set A is 3. If Set A has n elements, then, the total number of reflexive relations is given by the Reflexive Relations Formula N = 2n(n-1) Here, n = 3. Substituting the value, we get N = 23(3-1) = 29-3 = 26 = 64 Therefore, the total number of reflexive relations from set A to A is 64. |
Types of Reflexive Relations
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Some important types of Reflexive Relations are listed below:
- Anti-Reflexive Relation: Relation R defined on a Set A is considered an anti-reflexive relation if no element of A is related to itself. It is represented as (a, a) ∉ R for every a ∈ A. It is also referred to as Irreflexive Relation.
- Co-reflexive Relation: Relation R defined on a Set A is considered a co-reflexive relation if (a, b) ∈ R ⇒ a = b for all a, b ∈ A.
- Quasi-reflexive Relation: Relation R defined on a Set A is considered a quasi-reflexive relation if (a, b) ∈ R ⇒ (a, a) ∈ R and (b, b) ∈ R for all a, b ∈ A.
- Left Quasi-reflexive Relation: Relation R defined on a Set A is considered a left quasi-reflexive relation if (a, b) ∈ R ⇒ (a, a) ∈ R for all a, b ∈ A.
- Right Quasi-reflexive Relation: Relation R defined on a Set A is considered a right quasi-reflexive relation if (a, b) ∈ R ⇒ (b, b) ∈ R for all a, b ∈ A.
Reflexive Relations Solved Examples
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Here are some solved examples on Reflexive Relation for better concept clarity:
Example 1: A relation R is defined on the set of integers Z as aRb if and only if 3a + 4b is divisible by 7. Check if R is a reflexive relation.
Solution: For a ∈ Z, 3a + 4a = 7a which is clearly divisible by 7.
⇒ aRa
Since a is an arbitrary element of Z, therefore (a, a) ∈ R for all a ∈ Z.
Hence, R is a reflexive relation.
Thus, R is defined on Z as aRb if and only if 3a + 4b is divisible by 7 is a reflexive relation.
Example 2: A relation R is defined on the set of natural numbers N as aRb if and only if a ≥ b. Check if R is a reflexive relation.
Solution: For a ∈ N, a = a, which satisfies a ≥ a for every a ∈ N.
⇒ aRa
Since a is an arbitrary element of N, therefore (a, a) ∈ R for all a ∈ N
Hence, R is a reflexive relation.
Thus, R is defined on N as aRb if and only if a ≥ b is a reflexive relation.
Check More:
| Related Topics | ||
|---|---|---|
| Function Notation Formula | Domain and Range Relations | Period of a Function |
| Antisymmetric Relation | Polynomial Functions | Linear Functions |
Things to Remember
- Reflexive Relation is a relation in which every element of the set is related to itself.
- It possesses a reflexive property and is said to hold reflexivity.
- It is one of the three relations describing equivalence relations along with Symmetric and Transitive relations.
- A relation is said to be a Reflexive Relation if (a, a) ∈ R ∀ a ∈ A.
- Number of Reflexive Relations on a set with the 'n' number of elements is calculated by the Reflexive Relation Formula.
- Reflexive Relation Formula is N = 2n(n-1), where N is the number of reflexive relations and n is the number of elements in the set.
Previous Year Questions
- Let R be a reflexive relation on a finite set A having n-elements...
- (x − 1) (x2 − 5x + 7) < (x −,1) then x belongs to… (BITSAT – 2007)
- Consider the following lists… (AP EAPCET)
- If a + π/2 < 2 tan−1… (KCET – 2019)
- The domain and range of relation R = {x,y}/x,y...
- If A = {x | x∈N, x ≤ 5}, B = {x | x∈Z, x2 − 5x + 6 = 0}… (KCET – 2019)
- f : R → R and g : [0, ∞) → R is defined by… (KCET – 2019)
- cos[2sin−13/4 + cos−13/4]… (KCET – 2019)
- On the set of positive rationals, a binary operation… (KCET – 2019)
- R is a relation on N given by R = {(x,y) ∣ 4x… (KCET - 2008)
Sample Questions
Ques. A relation R is defined on Set A, a Set of All Integers, by “x R y if and only if 2x + 3y is divisible by 5”, for all x, y ∈ A. Check whether R is a reflexive relation on A or not. (2 Marks)
Ans. Assume that x ∈ A.
Now, 2x + 3x = 5x, which can be divided by 5.
It means xRx holds for all ‘x’ in A
Thus, R is reflexive relation.
Ques. A relation R is defined on Set N, a Set of all Real Numbers by ‘a R b’ if and only if |a-b| ≤ b, for a, b ∈ N. Prove that R is not Reflexive Relation. (2 Marks)
Ans. We know that a relationship is not reflexive if a = -2 ∈ R
However, |a – a| = 0 which is not less than -2(= a).
Thus, the relation R is not reflexive.
Ques. A relation R is defined on the Set A by “x R y if x – y is divisible by 5” for x, y ∈ A. Check whether R is a reflexive relation on Set A or not. (2 Marks)
Ans. Assume that x ∈ A.
Then, x – x is divisible by 5.
It means x R x holds for all x in A
Thus, R is a reflexive relation.
Ques. How to Calculate the Number of Reflexive Relations on a Set? (3 Marks)
Ans. The number of reflexive relations on a set with the ‘n’ number of elements can be calculated using the Reflexive Relation Formula.
Reflexive Relation Formula is given as
N = 2n(n-1)
Where
- N denotes the number of reflexive relations.
- n denotes the number of elements in the set.
Ques. Consider 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. Prove that R is a reflexive relation on set A. (2 Marks)
Ans. Assume that x ∈ A.
Thus, x + 3x = 4x is divisible by 4.
As x R x holds for all x in A.
Hence, R is a reflexive relation.
Ques. State the Difference between a Reflexive Relation and an Identity Relation. (3 Marks)
Ans. Relation R is considered to be an Identity Relation if R relates every element of a set to itself only. In simpler terms, an identity relation cannot relate an element to any element other than itself.
Relation R is reflexive relation if R relates every element of a set to itself. In simpler terms, a reflexive relation can relate an element to other elements along with relating the element with itself.
Ques. 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. (2 Marks)
Ans. 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 relation.
Ques. Is Reflexive Relation also Symmetric? (1 Mark)
Ans. Reflexive Relation may or may not be symmetric in nature. For instance, Relation R = {(a, a), (b, b), (c, c), (a, b), (a, c), (c, a)} defined on Set A = {a, b, c} is reflexive but not symmetric as (a, b) ∈ R but (b, a) ∉ R.
Ques. 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. (2 Marks)
Ans. 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.
Ques. 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. (2 Marks)
Ans. 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.
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