Difference Between Antisymmetric, Asymmetric, and Symmetric Relations
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 Explained with Examples
1. What is an antisymmetric relation in mathematics?
An antisymmetric relation R on a set A is a binary relation where, if (a, b) ∈ R and (b, a) ∈ R, then a must equal b. In simpler terms, if two distinct elements are related in both directions, the relation is not antisymmetric. This is a key concept in set theory and discrete mathematics.
2. Can you give an example of an antisymmetric relation?
The "less than or equal to" (≤) relation on the set of real numbers is antisymmetric. If a ≤ b and b ≤ a, then a = b. Another example is the subset relation (⊆) on a collection of sets. If A ⊆ B and B ⊆ A, then A = B.
3. How do you check if a relation is antisymmetric?
To verify antisymmetry, examine all pairs (a, b) in the relation. If you find a pair (a, b) and its reverse (b, a), both present in the relation, ensure a = b. If this holds true for all such pairs, the relation is antisymmetric. Otherwise, it's not.
4. What is the difference between antisymmetric and asymmetric relations?
An asymmetric relation R means that if (a, b) ∈ R, then (b, a) ∉ R. An antisymmetric relation allows (a, b) ∈ R and (b, a) ∈ R only if a = b. An asymmetric relation is a stricter condition; it's a subset of antisymmetric relations. The 'less than' (<) relation is asymmetric (and therefore antisymmetric), while 'less than or equal to' (≤) is antisymmetric but not asymmetric.
5. Can a relation be both symmetric and antisymmetric?
Yes. The equality relation (=) is both symmetric and antisymmetric. If a = b, then b = a (symmetric), and if a = b and b = a, then a = b (antisymmetric). The empty relation is also both symmetric and antisymmetric.
6. Is the empty relation antisymmetric?
Yes, the empty relation is considered antisymmetric because the condition for antisymmetry (if (a, b) ∈ R and (b, a) ∈ R, then a = b) is vacuously true. There are no pairs in the relation to violate the condition.
7. How is an antisymmetric relation represented in a matrix?
In an adjacency matrix representing an antisymmetric relation, if there's a 1 at position (i, j) indicating a relationship between elements i and j, and there's also a 1 at position (j, i), then i must equal j. Otherwise, only one of (i, j) and (j, i) can contain a 1.
8. Can a relation be antisymmetric but not transitive?
Yes. Antisymmetry and transitivity are independent properties. Consider the relation R = {(1, 1), (2, 2), (1, 2)} on the set {1, 2}. R is antisymmetric, but it is not transitive because (1, 2) and (2, 2) are in R, but (1, 2) is not in R (this is a simplified example and could be more complex depending on the set).
9. Do all partial orders require antisymmetry?
Yes, antisymmetry is a fundamental requirement for a partial order. A partial order is a relation that is reflexive, antisymmetric, and transitive. Without antisymmetry, it wouldn't be a partial order.
10. What are some real-world examples of antisymmetric relations?
Examples include: the "is a ancestor of" relation (in a family tree), the "is a subset of" relation among sets, and the "is less than or equal to" relation among numbers.
11. How does antisymmetry relate to partial ordering?
Antisymmetry is one of the three defining properties of a partial order (reflexivity and transitivity being the others). A partial order is a relation that is reflexive, antisymmetric, and transitive. This makes antisymmetry crucial in defining structures used in various areas of mathematics and computer science.
12. What is the difference between a symmetric relation and an antisymmetric relation?
A symmetric relation means if (a, b) is in the relation, then (b, a) must also be in the relation. An antisymmetric relation means that if both (a, b) and (b, a) are in the relation, then a and b must be equal. They are distinct concepts with opposing implications regarding the relationships between pairs of elements.
