How to Identify If a Relation Is Transitive or Not?
Mastering Transitive Relations is key for understanding complex mathematical structures, sets, and logic reasoning asked in board and entrance exams. Recognizing how relations connect elements can boost your marks and problem-solving speed, especially when comparing types like reflexive and symmetric. Explore their patterns for exam mastery!
Formula Used in Transitive Relations
The standard formula is: \( \text{For all } a, b, c \in A,\, (a\,Rb \text{ and } b\,Rc) \implies a\,Rc \)
Here’s a helpful table to understand Transitive Relations more clearly:
Transitive Relations Table
| Relation | Set Example | Is Transitive? |
|---|---|---|
| is less than (<) | (1,2), (2,3) ⇒ (1,3) | Yes |
| is a friend of | (A,B), (B,C) ⇒ (A,C)? | No (not always) |
| is equal to (=) | (4,4), (4,4) ⇒ (4,4) | Yes |
| is sibling of | (P,Q), (Q,R) ⇒ (P,R) | Yes |
This table shows how the pattern of Transitive Relations appears regularly in real examples and where it might not hold.
Worked Example – Solving a Transitive Relation Problem
1. Let set \( A = \{1,2,3\} \), and define relation \( R = \{(1,2), (2,3), (1,3)\} \).2. Check: If (1,2) and (2,3) are both in \( R \), then is (1,3) also in \( R \)?
3. Since for all such pairs, whenever (a, b) and (b, c) are in \( R \), (a, c) is too—so, \( R \) is a transitive relation.
Final Answer: The given relation is transitive.
Practice Problems
- State if the relation “is a factor of” on the set {2, 4, 8} is transitive.
- On set B = {a, b, c}, check if \( R = \{(a,b), (b,c)\} \) is transitive.
- Is “is a sibling of” transitive in every situation?
- Give an example of a relation that is not transitive.
- Compare Reflexive, Symmetric, and Transitive relations.
Common Mistakes to Avoid
- Assuming all relations are transitive without checking each pair.
- Mixing up transitive relations with symmetric or reflexive ones—use definitions precisely.
- Ignoring missing “middle link” pairs (b, c) in testing transitivity.
- Forgetting a relation can be transitive even if it's not symmetric or reflexive; check each property separately.
- Not using step-by-step logic for set-based relations or skipping pairs in exam questions.
Real-World Applications
Transitive relations are used in mathematics, computer science (like databases and closure property), geometry (as with congruence), and logic systems. Understanding these helps with grasping equivalence relations and advanced set operations, which are fundamental for programming, proofs, and relational databases. Vedantu can guide you in spotting such connections in diverse topics.
We explored the idea of Transitive Relations, their formula, examples, key differences, and real-world uses. Use step-by-step checking for all relation problems, stay clear on definitions, and practice regularly. For more on types and properties of relations, see Relations and Its Types on Vedantu.
FAQs on Understanding Transitive and Intransitive Relations with Examples
1. What is a transitive relation with an example?
Transitive relation is a property of a relation where if element a is related to b, and b is related to c, then a is also related to c. For example, in the set of numbers, the relation "is greater than" (>) is transitive: if 5 > 3 and 3 > 1, then 5 > 1.
2. What is an intransitive relation?
An intransitive relation is a relation in which, when element a is related to b and b is related to c, it does not guarantee that a is related to c. For example, the relation "is parent of" is intransitive. If A is the parent of B and B is the parent of C, A is not the parent of C.
3. How to know if a relation is transitive?
To determine if a relation R on set A is transitive, check:
- For every pair where (a, b) ∈ R and (b, c) ∈ R, also check if (a, c) ∈ R.
If this holds for all elements in the relation, then R is transitive.
4. What is an example of a relation that is not transitive?
A relation is not transitive if there exist elements a, b, and c such that (a, b) and (b, c) are in the relation, but (a, c) is not. For example, the relation "is the brother of" is not transitive: If A is the brother of B and B is the brother of C, A is not necessarily the brother of C.
5. What is the transitive property?
The transitive property states that for any elements a, b, and c: if a is related to b, and b is related to c, then a is also related to c. This is common in mathematics, such as: if a = b and b = c, then a = c.
6. What is a symmetric relation?
A symmetric relation is a relation where if element a is related to b, then b is also related to a. For example, the relation "is sibling of" is symmetric: if A is a sibling of B, then B is also a sibling of A.
7. What is a reflexive relation?
A reflexive relation is a relation on a set where every element is related to itself. That is, for a relation R on set A, (a, a) ∈ R for every a ∈ A. For instance, the relation "is equal to" ( = ) is reflexive, since every number is equal to itself.
8. What is the transitive relation formula in sets?
In set theory, a relation R on set A is transitive if for all a, b, c ∈ A:
If (a, b) ∈ R and (b, c) ∈ R, then (a, c) ∈ R.
9. What is an equivalence relation?
An equivalence relation is a relation that is reflexive, symmetric, and transitive. An example is the relation "has the same remainder when divided by 3" on integers.
10. What is transitive closure?
Transitive closure of a relation R is the smallest transitive relation that contains R. It adds the minimum needed pairs so that if (a, b) and (b, c) are in the relation, (a, c) is also included.
11. What is the transitive property of congruence?
The transitive property of congruence says that if two geometric figures (such as segments or angles) are each congruent to a third figure, then they are congruent to each other. For example, if AB ≅ CD and CD ≅ EF, then AB ≅ EF.
12. How are transitive, symmetric, and reflexive relations different?
Transitive, symmetric, and reflexive relations are types of relations on sets:
- Reflexive: Every element relates to itself.
- Symmetric: If a relates to b, then b relates to a.
- Transitive: If a relates to b and b relates to c, then a relates to c. A relation may have one, two, all, or none of these properties.
