Symmetric vs Asymmetric Relations: Key Differences and Examples
Symmetric Relations are part of set theory and discrete mathematics, and mastering them is essential for board exams and competitive tests like JEE. Understanding how these relations work helps students solve problems about relations and functions in exam situations, as well as draws connections to real-life pairings and social networks.
Formula Used in Symmetric Relations
The standard formula is: \( N = 2^{\frac{n(n+1)}{2}} \), where N is the number of symmetric relations on a set of n elements.
Here’s a helpful table to understand symmetric relations more clearly:
Symmetric Relations Table
| Ordered Pair | Is (b, a) also in relation? | Symmetric? |
|---|---|---|
| (2, 3) | Yes: (3, 2) included | Yes |
| (4, 5) | No: (5, 4) missing | No |
This table shows how the pattern of symmetric relations appears in real problem sets, quickly signaling whether a relation satisfies symmetry.
Worked Example – Solving a Problem
1. You have a set A = {1, 2, 3}. Relation R = {(1,1), (1,2), (2,1), (2,3)}. Is R a symmetric relation?Step 1: Examine if for each (a, b) in R, the pair (b, a) is also in R.
(1,1) — Its reverse (1,1) is clearly present.
(1,2) — Is (2,1) in R? Yes.
(2,1) — Is (1,2) in R? Yes.
(2,3) — Is (3,2) in R? No.
2. Since (2,3) is in R but (3,2) is missing, R is not symmetric.
Final Answer: The relation R is not symmetric because not every (a, b) is matched by (b, a).
Practice Problems
- Given A = {a, b}, list all symmetric relations on A.
- If (a, b) and (b, a) are both in R, must R be symmetric? Explain.
- Find the number of symmetric relations possible on set A = {1, 2, 3}.
- For the relation “is a sibling of,” explain why it is symmetric, and compare to “is a parent of.”
Common Mistakes to Avoid
- Assuming a relation is symmetric if only some pairs have their reverse included.
- Confusing symmetric and reflexive: not all symmetric relations are reflexive, and vice versa. See examples on Reflexive Relation.
- Mixing up symmetric with antisymmetric. Visit Antisymmetric Relation for a detailed comparison.
Real-World Applications
Symmetric relations appear in everyday contexts like “is a friend of,” “has the same birthday as,” or “lives in the same city as.” These concepts are used in computer science, networking, social media, and database design. Vedantu highlights such real-world links to deepen student understanding.
Page Summary
We explored the topic of symmetric relations, discussing the definition, formula, pattern recognition, worked examples, and key mistakes to avoid. Review interconnected topics like Relations and Functions and Equivalence Relation to master this concept for exams and real life. Keep practicing with Vedantu resources for more confidence in set theory!
FAQs on What Is a Symmetric Relation? Definition, Properties & Examples
1. What is a symmetric relation with example?
Symmetric relation is a relation in which if element a is related to element b, then b is also related to a. For example, in the set A = {1, 2, 3}, if (1, 2) belongs to relation R, then (2, 1) must also belong to R for it to be symmetric. An example is the relation R = {(1, 2), (2, 1), (2, 3), (3, 2)}.
2. What is symmetric and asymmetric relation?
Symmetric relation means if (a, b) is in the relation, then (b, a) must also be there. In contrast, an asymmetric relation means if (a, b) is in the relation, then (b, a) cannot be in the relation. For example, ‘is sibling of’ is symmetric, but ‘is father of’ is asymmetric.
3. What is the symmetric relation of 1 2 3 4?
For set A = {1, 2, 3, 4}, a symmetric relation includes pairs such that if (a, b) is present, (b, a) is also present. One example is R = {(1,2), (2,1), (3,3), (4,4)}. All such pairs satisfy the symmetric property.
4. What is an example of a symmetric relation?
An example of a symmetric relation is 'is married to'. If A is married to B, then B is also married to A. For a mathematical example: In A = {1, 2}, the relation R = {(1, 2), (2, 1)} is symmetric.
5. What is the formula for symmetric relations?
For a set with n elements, the number of symmetric relations is 2n(n+1)/2. This is because for each pair (a, b) where a ≠ b, you must include both (a, b) and (b, a) or neither.
6. What is a transitive relation?
A transitive relation on set A means that if (a, b) and (b, c) are in the relation, then (a, c) must also be in the relation. For example, ‘is an ancestor of’ is transitive.
7. What is a reflexive relation?
A reflexive relation on set A means every element is related to itself. That is, for all a ∈ A, (a, a) must be included in the relation.
8. What are antisymmetric relations?
Antisymmetric relation means if (a, b) and (b, a) are in the relation, then a must equal b. For example, ‘less than or equal to’ (≤) is antisymmetric, because if a ≤ b and b ≤ a, then a = b.
9. What is symmetric relation in discrete mathematics?
In discrete mathematics, a symmetric relation over a set A is a relation R where if (a, b) ∈ R, then (b, a) ∈ R for all a, b ∈ A.
10. Indicate which relations in set A are defined as symmetric.
A relation in set A is defined as symmetric if for every (a, b) in the relation, the pair (b, a) is also present. This must be true for all pairs in the relation.
11. What is symmetric relational responding?
In behavioral science, symmetric relational responding refers to the ability to derive that if relation A stands in relation to B, then B stands in the same relation to A. For example, if ‘A is the same as B’, then ‘B is the same as A’.
12. What is the meaning of symmetric relation in Hindi?
हिंदी में, समानुपाती संबंध (Symmetric Relation) वह संयोजन है जिसमें यदि (a, b) संबंध में है, तो (b, a) भी संबंध में होना चाहिए। उदाहरण के लिए, यदि 1, 2 संबंधित हैं तो 2, 1 भी संबंधित होंगे।
