Types of Relations

Types of Relations

A relation describes a connection between the elements of two sets. Relations are defined using ordered pairs and Cartesian products.

यदि A और B दो non-empty sets हैं, तो A और B के बीच relation, A × B के किसी subset के रूप में defined किया जाता है।

1. Ordered Pair

An ordered pair is written as (a, b), where a is the first component and b is the second component.

दो ordered pairs तभी equal होते हैं जब उनके corresponding components equal हों:

(a, b) = (c, d) ⟺ a = c and b = d

Example

If:

(x, 5) = (2, y)

Comparing the corresponding components:

x = 2 and y = 5

2. Cartesian Product

For two non-empty sets A and B, the Cartesian product A × B is the set of all ordered pairs (a, b), where a ∈ A and b ∈ B.

A × B = {(a, b) : a ∈ A, b ∈ B}

Example

Let:

A = {1, 2}

B = {a, b}

Then:

A × B = {(1,a), (1,b), (2,a), (2,b)}

Similarly:

B × A = {(a,1), (a,2), (b,1), (b,2)}

3. Definition of Relation

A relation R from a non-empty set A to a non-empty set B is a subset of A × B.

Therefore:

R ⊆ A × B

तो R, A से B में एक relation है।

Example

Let:

A = {1, 2, 3}

B = {2, 4, 6}

Suppose:

R = {(1,2), (2,4), (3,6)}

Since every ordered pair of R belongs to A × B:

R ⊆ A × B

Hence, R is a relation from A to B.

4. Domain, Codomain and Range

Suppose R is a relation from A to B.

Domain

The domain of R is the set of all first components of the ordered pairs in R.

Codomain

The set B is called the codomain of R.

Range

The range of R is the set of all second components that actually occur in the ordered pairs of R.

Example

Let:

A = {1, 2, 3, 4}

B = {a, b, c}

R = {(1,a), (2,b), (3,b)}

Then:

Domain = {1, 2, 3}

Codomain = {a, b, c}

Range = {a, b}

Types of Relations

Relations को उनकी properties के आधार पर अलग-अलग types में classify किया जाता है.

• Empty Relation
• Universal Relation
• Identity Relation
• Reflexive Relation
• Symmetric Relation
• Transitive Relation
• Equivalence Relation

5. Empty Relation

A relation R on a set A is called an empty relation if no element of A is related to any element of A.

Therefore:

R = ∅

Example

Let A = {1, 2, 3} and:

R = ∅

Since R contains no ordered pair, R is an empty relation on A.

6. Universal Relation

A relation R on a set A is called a universal relation if every element of A is related to every element of A.

Therefore:

R = A × A

Example

Let A = {1, 2}.

Then:

A × A = {(1,1), (1,2), (2,1), (2,2)}

If:

R = {(1,1), (1,2), (2,1), (2,2)}

then R is the universal relation on A.

7. Identity Relation

A relation R on a set A is called an identity relation if every element is related only to itself.

It is represented by:

I_A = {(a,a) : a ∈ A}

Example

Let:

A = {1, 2, 3}

Then:

I_A = {(1,1), (2,2), (3,3)}

इस relation में हर element केवल अपने आप से related है.

8. Reflexive Relation

A relation R on a set A is called reflexive if every element of A is related to itself.

Mathematically:

(a,a) ∈ R for every a ∈ A

How to Check?

यदि A = {1, 2, 3} है, तो reflexive relation के लिए ये pairs compulsory हैं:

(1,1), (2,2), (3,3)

इनमें से कोई भी pair missing है, तो relation reflexive नहीं होगा.

Example

Let:

A = {1, 2, 3}

R = {(1,1), (2,2), (3,3), (1,2)}

Since (1,1), (2,2) and (3,3) all belong to R, R is reflexive.

Non-Example

Let:

R = {(1,1), (2,2), (1,2)}

Here (3,3) is missing.

Therefore R is not reflexive.

9. Symmetric Relation

A relation R on a set A is called symmetric if:

(a,b) ∈ R ⇒ (b,a) ∈ R

अर्थात यदि a का relation b से है, तो b का relation भी a से होना चाहिए.

Example

Let:

R = {(1,1), (2,2), (1,2), (2,1)}

Here:

(1,2) ∈ R and (2,1) ∈ R.

Therefore R is symmetric.

Non-Example

Let:

R = {(1,2), (2,3)}

Here (1,2) ∈ R but (2,1) ∉ R.

Therefore R is not symmetric.

10. Transitive Relation

A relation R on a set A is called transitive if:

(a,b) ∈ R and (b,c) ∈ R ⇒ (a,c) ∈ R

इसमें तीन elements के बीच connection check किया जाता है.

Example

Let:

R = {(1,2), (2,3), (1,3)}

Here:

(1,2) ∈ R and (2,3) ∈ R.

Therefore (1,3) must be present.

Since (1,3) ∈ R, the condition is satisfied.

Hence R is transitive.

Non-Example

Let:

R = {(1,2), (2,3)}

Here:

(1,2) ∈ R and (2,3) ∈ R, but (1,3) ∉ R.

Therefore R is not transitive.

11. Equivalence Relation

A relation R on a set A is called an equivalence relation if it is reflexive, symmetric and transitive.

Equivalence Relation = Reflexive + Symmetric + Transitive

Example

Let A = {1, 2, 3} and:

R = {(1,1), (2,2), (3,3), (1,2), (2,1)}

Step 1: Check Reflexivity

(1,1), (2,2) and (3,3) are present in R.

Therefore R is reflexive.

Step 2: Check Symmetry

(1,2) ∈ R and (2,1) ∈ R.

Therefore R is symmetric.

Step 3: Check Transitivity

The required combinations satisfy the transitive condition.

Therefore R is transitive.

Hence R is an equivalence relation.

12. Important Difference Between Relations

Type	Condition
Empty Relation	R = ∅
Universal Relation	R = A × A
Identity Relation	IA = {(a,a) : a ∈ A}
Reflexive Relation	(a,a) ∈ R for every a ∈ A
Symmetric Relation	(a,b) ∈ R ⇒ (b,a) ∈ R
Transitive Relation	(a,b), (b,c) ∈ R ⇒ (a,c) ∈ R
Equivalence Relation	Reflexive + Symmetric + Transitive

13. Solved Example: Identify the Type of Relation

Let A = {1,2,3} and:

R = {(1,1),(2,2),(3,3),(1,2),(2,1)}

Reflexive: All required pairs (1,1), (2,2), (3,3) are present.

Symmetric: (1,2) के साथ (2,1) भी present है.

Transitive: Required combinations satisfy the transitive condition.

Therefore R is an equivalence relation.

14. Solved Example: Relation Defined by Divisibility

Let A = {1,2,3,4} and define R by:

a R b if a divides b.

Check whether R is reflexive, symmetric and transitive.

Step 1: Reflexive

Every number divides itself:

1 | 1, 2 | 2, 3 | 3, 4 | 4.

Therefore R is reflexive.

Step 2: Symmetric

2 divides 4, but 4 does not divide 2.

Therefore R is not symmetric.

Step 3: Transitive

If a divides b and b divides c, then a divides c.

Therefore R is transitive.

Hence R is reflexive and transitive but not symmetric.

15. Solved Example: Relation Defined by Equality

On a set A, define:

a R b if a = b.

This relation contains only pairs of the form (a,a).

Therefore it is the identity relation.

Identity relation is reflexive, symmetric and transitive.

Hence it is an equivalence relation.

16. How to Solve Questions on Properties of Relations

जब किसी relation की properties check करने को कहा जाए, तो एक fixed order follow करना आसान रहता है:

1. Reflexive: सभी (a,a) pairs check करें.
2. Symmetric: हर (a,b) के लिए (b,a) check करें.
3. Transitive: (a,b) और (b,c) मिलने पर (a,c) check करें.
4. अगर तीनों properties satisfy हों, तो relation equivalence relation है.

17. Practice Set – Basic

1. Let A = {1,2,3}. Write A × A.
2. Let A = {1,2} and B = {a,b,c}. Find A × B.
3. Find the domain and range of R = {(1,2),(2,4),(3,6)}.
4. Write the identity relation on A = {1,2,3,4}.
5. Give an example of an empty relation on A = {1,2}.
6. Give an example of a universal relation on A = {1,2}.
7. Check whether R = {(1,1),(2,2),(1,2),(2,1)} is reflexive.
8. Check whether R = {(1,2),(2,1),(2,3),(3,2)} is symmetric.
9. Check whether R = {(1,2),(2,3),(1,3)} is transitive.
10. State the three conditions required for an equivalence relation.

18. Practice Set – Identify the Property

1. Let A = {1,2,3} and R = {(1,1),(2,2),(3,3)}. Identify the type of relation.

2. Let A = {1,2} and R = {(1,1),(1,2),(2,1),(2,2)}. Identify the type of relation.

3. Let A = {1,2,3} and R = {(1,1),(2,2),(3,3),(1,2)}. Is R reflexive?

4. Let A = {1,2,3} and R = {(1,2),(2,1),(2,3),(3,2)}. Is R symmetric?

5. Let A = {1,2,3} and R = {(1,2),(2,3),(1,3)}. Is R transitive?

19. True or False

1. Every identity relation is reflexive.
2. Every universal relation is reflexive.
3. Every empty relation is reflexive on a non-empty set.
4. Every equivalence relation is symmetric.
5. Every equivalence relation is transitive.
6. Every symmetric relation is reflexive.
7. Every reflexive relation is symmetric.
8. Range of a relation is always a subset of its codomain.
9. Generally, A × B = B × A.
10. An equivalence relation must be reflexive, symmetric and transitive.

Answers:

1. True   2. True   3. False   4. True   5. True   6. False   7. False   8. True   9. False   10. True

20. Multiple Choice Questions

1. A relation R from A to B is a subset of:

   (A) A   (B) B   (C) A × B   (D) A ∪ B

   Answer: (C) A × B

2. If A = {1,2} and B = {a,b}, then the number of elements in A × B is:

   (A) 2   (B) 3   (C) 4   (D) 5

   Answer: (C) 4

3. The domain of R = {(1,a),(2,b),(3,c)} is:

   (A) {a,b,c}   (B) {1,2,3}   (C) {1,a}   (D) {2,b}

   Answer: (B) {1,2,3}

4. A relation R on A is reflexive if:

   (A) (a,b) ∈ R ⇒ (b,a) ∈ R

   (B) (a,a) ∈ R for every a ∈ A

   (C) R = A × A

   (D) R = ∅

   Answer: (B)

5. A relation R is symmetric if:

   (A) (a,a) ∈ R

   (B) (a,b) ∈ R ⇒ (b,a) ∈ R

   (C) R = A × A

   (D) R = ∅

   Answer: (B)

6. A relation R is transitive if:

   (A) (a,b),(b,c) ∈ R ⇒ (a,c) ∈ R

   (B) (a,b) ∈ R ⇒ (b,a) ∈ R

   (C) (a,a) ∈ R

   (D) R = A × A

   Answer: (A)

7. An equivalence relation is:

   (A) Reflexive only

   (B) Symmetric only

   (C) Transitive only

   (D) Reflexive, symmetric and transitive

   Answer: (D)

8. The identity relation on A is:

   (A) A × A

   (B) ∅

   (C) {(a,a) : a ∈ A}

   (D) A ∪ A

   Answer: (C)

9. The universal relation on A is:

   (A) ∅   (B) A × A   (C) A   (D) {(a,a)}

   Answer: (B)

10. If R is an equivalence relation, which properties must it satisfy?

    (A) Reflexive   (B) Symmetric   (C) Transitive   (D) All of these

    Answer: (D) All of these

21. Fill in the Blanks

1. A relation from A to B is a subset of ________.
2. The set of first components of a relation is called its ________.
3. The set of actual second components is called the ________.
4. A relation containing no ordered pair is called an ________ relation.
5. A relation R on A is universal if R = ________.
6. A relation R is reflexive if ________ belongs to R for every a ∈ A.
7. A symmetric relation satisfies (a,b) ∈ R ⇒ ________.
8. A transitive relation satisfies (a,b),(b,c) ∈ R ⇒ ________.
9. An equivalence relation is reflexive, symmetric and ________.
10. I_A = {(a,a) : a ∈ A} is called the ________ relation.

Answers:

1. A × B
2. Domain
3. Range
4. Empty
5. A × A
6. (a,a)
7. (b,a) ∈ R
8. (a,c) ∈ R
9. Transitive
10. Identity

22. Important Board Exam Points

23. Quick Revision

Concept	Key Point
Relation	Subset of A × B
Domain	First components
Codomain	Target set B
Range	Actual second components
Empty Relation	R = ∅
Universal Relation	R = A × A
Identity Relation	IA = {(a,a) : a ∈ A}
Reflexive	(a,a) ∈ R for every a ∈ A
Symmetric	(a,b) ∈ R ⇒ (b,a) ∈ R
Transitive	(a,b),(b,c) ∈ R ⇒ (a,c) ∈ R
Equivalence	Reflexive + Symmetric + Transitive