Symmetric Relation definition properties and solved 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 Understanding Symmetric Relations in Set Theory
1. What is a symmetric relation in mathematics?
A symmetric relation is a relation R on a set A in which if (a, b) ∈ R, then (b, a) ∈ R for all a, b ∈ A. In simple terms, whenever one element is related to another, the reverse relation must also hold.
- Condition: If aRb, then bRa
- Applies to relations defined on the same set
- Order of elements is reversible
2. How do you check if a relation is symmetric?
To check if a relation is symmetric, verify that for every ordered pair (a, b) in R, the pair (b, a) is also in R.
- Step 1: List all ordered pairs in the relation.
- Step 2: For each (a, b), check whether (b, a) exists.
- Step 3: If any reverse pair is missing, the relation is not symmetric.
3. Can you give an example of a symmetric relation?
An example of a symmetric relation is the relation “is equal to” on real numbers. If a = b, then automatically b = a.
- Let A = {1, 2, 3}
- R = {(1,1), (2,2), (3,3), (1,2), (2,1)}
4. What is the difference between symmetric and reflexive relations?
The difference is that a symmetric relation requires reversibility of pairs, while a reflexive relation requires every element to relate to itself.
- Symmetric: If (a, b) ∈ R, then (b, a) ∈ R
- Reflexive: (a, a) ∈ R for every a in A
5. Is every symmetric relation also transitive?
No, a symmetric relation is not necessarily transitive. Symmetry only ensures reversibility, while transitivity requires chaining of relations.
- Transitive condition: If (a, b) ∈ R and (b, c) ∈ R, then (a, c) ∈ R
6. What is the symmetric closure of a relation?
The symmetric closure of a relation R is the smallest symmetric relation that contains R. It is obtained by adding (b, a) whenever (a, b) is in R.
- Formula: R ∪ R⁻¹
- Where R⁻¹ = {(b, a) | (a, b) ∈ R}
7. How is a symmetric relation represented in a matrix?
A relation is symmetric if its relation matrix is symmetric about the main diagonal. This means the matrix satisfies M = Mᵀ.
- If entry (i, j) = 1, then entry (j, i) must also be 1.
- The matrix is equal to its transpose.
8. How do you represent a symmetric relation using a digraph?
In a digraph, a relation is symmetric if every directed edge from a to b has a corresponding edge from b to a.
- If there is an arrow a → b, there must be b → a.
- Self-loops (a → a) are allowed but not required for symmetry.
9. What are some real-life examples of symmetric relations?
A real-life example of a symmetric relation is “is a sibling of” because if A is a sibling of B, then B is a sibling of A.
- “Is married to”
- “Has the same age as”
- “Is a classmate of”
10. Can a symmetric relation contain self-pairs like (a, a)?
Yes, a symmetric relation can contain self-pairs (a, a), but they are not required unless the relation is also reflexive. Symmetry only concerns pairs of distinct elements.
- If (a, a) ∈ R, symmetry is automatically satisfied for that pair.
- Self-pairs do not affect the symmetric condition.
