Definition formula properties and solved examples of antisymmetric relation
The concept of antisymmetric relation plays a key role in mathematics and is widely applicable to both real-life situations and exam scenarios. Understanding antisymmetric relations helps you solve important questions in sets, relations and functions, discrete mathematics, and logic topics in competitive exams like JEE and CBSE board exams.
What Is Antisymmetric Relation?
An antisymmetric relation is a special type of mathematical relation defined on a set. Formally, a relation \(R\) on a set \(A\) is called antisymmetric if, for all \(a, b \in A\), whenever both \((a, b)\) and \((b, a)\) belong to \(R\), then necessarily \(a = b\). You’ll find this concept applied in areas such as set theory, matrix representations, and computer science logic circuits.
Key Formula for Antisymmetric Relation
Here’s the standard formula: \( \forall a, b \in A, \ [(a, b) \in R \text{ and } (b, a) \in R] \implies a = b \)
Antisymmetric Relation Explained With Examples
| Relation Type | Example Set \(A = \{1,2,3\}\) | Is It Antisymmetric? | Why? |
|---|---|---|---|
| R1 = {(1,1),(2,2),(3,3)} | Self-loops only | Yes | No two distinct elements are mutually related |
| R2 = {(1,2),(2,1)} | 1 ↔ 2 | No | (1,2) and (2,1) exist, but 1 ≠ 2 |
| R3 = {(1,2),(2,2),(2,3)} | 1 → 2, 2 → 3, self-loop on 2 | Yes | For (a,b), (b,a) never both present unless a = b |
Antisymmetric Relation in Matrices and Graphs
Relations can be quickly checked for antisymmetry with matrices or graph diagrams. In a relation matrix for set \(A\), fill row \(i\), column \(j\) with “1” if \((i, j)\) exists in \(R\). The relation is antisymmetric if for every pair of different positions \((i, j)\) and \((j, i)\), both are not "1" unless \(i = j\).
- If both \(M_{ij} = 1\) and \(M_{ji} = 1\) and \(i \neq j\) –> Not antisymmetric.
In a directed graph, if there is a two-way arrow between different nodes, the relation is not antisymmetric.
Difference Between Antisymmetric, Asymmetric, and Symmetric Relations
| Property | Antisymmetric | Asymmetric | Symmetric |
|---|---|---|---|
| Formula/Definition | If (a,b) and (b,a) ∈ R, then a = b | If (a,b) ∈ R ⇒ (b,a) ∉ R | If (a,b) ∈ R ⇒ (b,a) ∈ R |
| (2,3) and (3,2) both in R? | Not allowed unless 2 = 3 | Never allowed | Always allowed |
| Can have self-loops? | Yes | No | Yes |
How to Check if a Relation Is Antisymmetric (Step-by-Step)
- List all pairs in the relation R.
- For each pair (a, b), check if (b, a) ≠ (a, b) and a ≠ b ALSO belongs to R.
- If you find any such pair where both (a, b) and (b, a) exist for a ≠ b, then relation is not antisymmetric.
- If no such pairs found, relation is antisymmetric.
Real-Life Example of Antisymmetric Relation
The “less than or equal to” (\(\leq\)) relation on real numbers is antisymmetric. If \(a \leq b\) and \(b \leq a\), it automatically means \(a = b\). Many relations in rankings, hierarchies, and data structures use antisymmetry for ordering and comparison.
Practice Questions – Try These Yourself
- Is the relation \(R = \{(1, 1), (2, 2), (1, 2)\}\) on set \(\{1,2\}\) antisymmetric?
- Does the “divides” relation (\(\mid\)) on integers form an antisymmetric relation?
- Given the matrix below, is the relation antisymmetric?
\[ \begin{pmatrix} 1 & 1 \\ 1 & 1 \\ \end{pmatrix} \]
- List a non-example of antisymmetric relation on set \(\{a, b\}\).
Frequent Errors and Misunderstandings
- Confusing antisymmetric with asymmetric (Remember: antisymmetric allows self-pairs; asymmetric does not.)
- Assuming that not symmetric = antisymmetric (not always true!)
- Overlooking self-loops—they are allowed in antisymmetric relations.
Relation to Other Concepts
The idea of antisymmetric relation connects closely with types of sets and reflexive relations. Mastering this helps you understand relation types such as partial ordering and equivalence, which are central in higher mathematics and computer science data structures.
Classroom Tip
A quick way to remember antisymmetric relation: “Forward and backward arrows are allowed at the same time only if they start and end at the same element.” Vedantu’s teachers often use simple matrices and diagrams to visualize this during live classes.
Summary Table: Key Points at a Glance
| Property | Antisymmetric? |
|---|---|
| Self-loops ((a, a)) allowed | Yes |
| Both (a, b) and (b, a) allowed? | Only if a = b |
| Relation “less than or equal to” (≤) | Antisymmetric |
| Relation “is sibling of” | Not antisymmetric |
| Easy test? | Look for two-way link between different elements |
- Antisymmetric is NOT the same as asymmetric.
- Reflexive and antisymmetric can exist together.
- Matrices and diagrams make checking antisymmetry faster.
We explored antisymmetric relation—from its definition and formula to distinctions, examples, and connections. Continue practicing with Vedantu to become confident in identifying antisymmetric relations and solving all kinds of relation problems in exams.
Further Reading and Related Topics
- Relations and Its Types – Covers all relations: symmetric, asymmetric, reflexive, etc.
- Symmetric and Skew Symmetric Matrix – Compare properties in matrix form.
- Reflexive Relation – Learn about another basic relation in set theory.
- Types of Sets – Foundation for all relation topics.
- Set Theory Symbols – For understanding notation in relation formulas.
FAQs on Antisymmetric Relation in Set Theory Explained
1. What is an antisymmetric relation?
An antisymmetric relation on a set A is a relation R such that if (a, b) ∈ R and (b, a) ∈ R, then a = b.
- This means two different elements cannot be related in both directions.
- If both (a, b) and (b, a) are present, the elements must be identical.
- It is a key property used in defining partial orders.
2. What is an example of an antisymmetric relation?
The relation ≤ (less than or equal to) on real numbers is an example of an antisymmetric relation.
- If a ≤ b and b ≤ a, then a = b.
- For example, if 5 ≤ 5 and 5 ≤ 5, then both numbers are equal.
- But if 3 ≤ 4, 4 ≤ 3 is false, so antisymmetry is not violated.
3. How do you check if a relation is antisymmetric?
To check if a relation R is antisymmetric, verify that whenever (a, b) and (b, a) are in R, then a = b.
- Step 1: List all ordered pairs in the relation.
- Step 2: Look for pairs (a, b) and (b, a).
- Step 3: If such pairs exist with a ≠ b, the relation is not antisymmetric.
4. What is the difference between symmetric and antisymmetric relations?
A symmetric relation requires that if (a, b) ∈ R, then (b, a) ∈ R, while an antisymmetric relation requires that if both (a, b) and (b, a) are in R, then a = b.
- Symmetric: Both directions must exist.
- Antisymmetric: Both directions can exist only if elements are equal.
- A relation can be both symmetric and antisymmetric only in special cases (like equality).
5. Can a relation be both symmetric and antisymmetric?
Yes, a relation can be both symmetric and antisymmetric if it only relates elements to themselves, such as the equality relation.
- For equality, (a, b) and (b, a) imply a = b.
- No distinct elements are related in both directions.
- Such relations are typically identity relations.
6. Is the equality relation antisymmetric?
Yes, the equality relation (=) is antisymmetric because if a = b and b = a, then a and b are the same element.
- It satisfies the antisymmetric condition naturally.
- It is also reflexive and symmetric.
- Therefore, equality is a special type of relation with multiple properties.
7. What is the formula or condition for an antisymmetric relation?
The formal condition for an antisymmetric relation R on set A is: if (a, b) ∈ R and (b, a) ∈ R, then a = b.
- Symbolically: (aRb ∧ bRa) ⇒ a = b.
- This must hold for all a, b ∈ A.
- This condition is essential in defining partially ordered sets (posets).
8. Is the “less than” relation antisymmetric?
Yes, the less than relation (<) is antisymmetric because it is impossible for a < b and b < a to both be true unless a = b, which cannot happen.
- If a < b, then b < a is false.
- The condition for antisymmetry is satisfied vacuously.
- Thus, < is an antisymmetric and transitive relation.
9. What is the role of antisymmetric relation in partial order?
An antisymmetric relation is one of the three properties required for a partial order.
- A partial order must be reflexive, antisymmetric, and transitive.
- Antisymmetry ensures that no two distinct elements precede each other.
- Example: The subset relation (⊆) forms a partially ordered set.
10. What is a common mistake when identifying antisymmetric relations?
A common mistake is confusing antisymmetric with not symmetric.
- Antisymmetric does not mean the opposite of symmetric.
- It only restricts cases where both (a, b) and (b, a) appear.
- If such pairs exist for distinct elements, the relation is not antisymmetric.
