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An antisymmetric relation is an important concept in mathematics, particularly in set theory and discrete mathematics. This type of relation is defined on a set, where for any two elements a and b, if both a is related to b and b is related to a, then a must be equal to b. Antisymmetric relations are significant in various mathematical disciplines and are essential in understanding order theory and equivalence relations. Important properties of antisymmetric relations include reflexivity, transitivity, and the uniqueness of elements in the relation. Understanding antisymmetric relations is crucial for solving problems in competitive exams like GATE, IIT-JEE, GRE, and other engineering and mathematical entrance tests. This Antisymmetric Relation blog also involves exploring conditions for a relation to be antisymmetric, graph representations of these relations, specific rules to identify them, and practical examples of antisymmetric relations to solidify the concept.
Table of Contents
What is Antisymmetric Relation?
An antisymmetric relation is a type of binary relation defined on a set that exhibits a specific characteristic involving pairs of elements. In simple terms, a relation R on a set A is called antisymmetric if for all pairs of elements a and b in A, whenever a is related to b (i.e., aRb) and b is related to a (i.e., bRa), it must be the case that a = b.
Important Terms:
- Binary Relation: A binary relation R on a set A is a collection of ordered pairs (a,b)(a, b)(a,b) where a and b are elements of A.
- Set A: A collection of distinct objects, considered as an object in its own right. For instance, A={1,2,3}.
- Ordered Pair: A pair of elements (a,b) where the order of elements matters, meaning (a,b) is different from (b,a) unless a = b.
- Relation R: A subset of the Cartesian product A × A, meaning R ⊆ A×A.
Formal Definition:
A relation R on a set A is antisymmetric if:
| ∀a, b∈A, if aRb and bRa then a = b |
Example:
Consider the set A = {1,2,3} and the relation R = {(1,1),(2,2),(3,3),(1,2)}. In this case, the relation is antisymmetric because there are no elements a and b such that a ≠ b and both (a,b) and (b,a) belong to R.
Properties of Antisymmetric Relation
Antisymmetric relations have several important properties and characteristics that make them significant in various areas of mathematics. Here are the important properties of Antisymmetric Relation:
1. Definition and Basic Property
- Property: A relation R on a set A is antisymmetric if ∀a, b∈A, whenever aRb and bRa, then a = b.
- Example: Let A = {1,2,3} and R = {(1,1),(2,2),(3,3),(1,2)}. Here, the relation R is antisymmetric because there are no pairs (a,b) and (b,a) in R where a ≠ b.
2. Reflexivity
- Property: An antisymmetric relation does not have to be reflexive, but if it is reflexive, it means every element is related to itself.
- Example: Consider the relation R on A = {1,2} where R = {(1,1),(2,2)}. This relation is both reflexive (every element is related to itself) and antisymmetric (there are no pairs (a,b) and (b,a) with a≠b).
3. Transitivity
- Property: Antisymmetric relations can be transitive. Transitivity means if aRb and bRc, then aRc.
- Example: Let A = {1,2,3} and R = {(1,2),(2,3),(1,3)}. This relation is antisymmetric (no pairs (a,b) and (b,a) with a≠b) and transitive (if 1R2 and 2R3, then 1R3).
4. Partial Order
- Property: A relation that is antisymmetric, reflexive, and transitive is called a partial order.
- Example: The relation ≤ (less than or equal to) on the set of integers Z is a partial order. For any integers a, b, and c:
- Antisymmetric: If a ≤ b and b ≤ a, then a = b.
- Reflexive: a ≤ a for any a.
- Transitive: If a ≤ b and b ≤ c, then a ≤ c.
5. Graph Representation
- Property: In graph theory, an antisymmetric relation can be represented as a directed graph where there are no two distinct nodes aaa and bbb with both a→b and b→a.
- Example: In the directed graph of the relation R = {(1,2),(2,3),(1,3)} on A = {1,2,3}, there are no edges going both ways between any pair of nodes, which illustrates antisymmetry.
6. Examples and Non-Examples
- Example: Consider the set A = {a,b} and the relation R = {(a,a),(b,b)}. This relation is antisymmetric because it does not contain any pairs (a,b) and (b,a) with a ≠ b.
- Non-Example: Let A = {1,2} and R = {(1,2),(2,1)}. This relation is not antisymmetric because 1R2 and 2R1 but 1≠2.
7. Connection with Other Relations
- Property: Antisymmetric relations are related to partial orders but are not necessarily equivalence relations.
- Example: The relation “is a subset of” (⊆) on the set of all subsets of a given set is antisymmetric. If A⊆B and B⊆A, then A=B. It is also a partial order because it is reflexive and transitive.
Also Read: All Perfect Cube Numbers
Conditions of Antisymmetric Relation
For a relation R on a set A to be considered antisymmetric, it must satisfy the following condition:
Antisymmetric Condition
A relation R on a set A is antisymmetric if for all elements a and b in A: If aRb and bRa, then a = b.
Conditions for Antisymmetric Relation
- Pairwise Condition:
- Condition: For any pairs (a,b) and (b,a) in the relation R, if both pairs exist, then aaa must be equal to b.
- Example: In the relation R = {(1,2),(2,1)} on the set A = {1,2}, the pairs (1,2) and (2,1) violate antisymmetry because 1≠2. Therefore, this relation is not antisymmetric.
- No Bidirectional Pairs:
- Condition: There should be no two distinct elements a and b in A such that both aRb and bRa hold true, unless a=b.
- Example: In the relation R = {(1,1),(2,2),(1,2)} on the set A = {1,2}, there are no instances where both (1,2) and (2,1) are present. Hence, the relation is antisymmetric.
- Irreflexivity and Reflexivity:
- Condition: While antisymmetry does not require reflexivity, if a relation is both reflexive and antisymmetric, then it is a partial order.
- Example: The “less than or equal to” relation (≤) on integers is both reflexive (every number is less than or equal to itself) and antisymmetric (if a ≤ b and b ≤ a).
- Graph Representation:
- Condition: In a directed graph representing the relation, there should be no edges going both ways between any two distinct nodes.
- Example: For the relation R = {(1,2),(2,3)} on A = {1,2,3}, the directed graph shows no bidirectional edges between different nodes, which supports antisymmetry.
- Relation with Other Properties:
- Condition: Antisymmetry alone does not imply reflexivity or transitivity. However, when combined with reflexivity and transitivity, the relation becomes a partial order.
- Example: The subset relation (⊆) on a set of subsets is antisymmetric, reflexive, and transitive, making it a partial order.
Rules of Antisymmetric Relation
Understanding the rules governing antisymmetric relations helps in identifying and working with such relations in different mathematical contexts. Here are the important rules of antisymmetric relations:
Rules of Antisymmetric Relations
- Rule of Non-Existence of Bidirectional Relations:
In an antisymmetric relation R on a set A, if (a,b) ∈ R and (b,a) ∈ R, then a must be equal to b.
- Example: If R = {(1,2),(2,1)} on the set {1,2}, this relation is not antisymmetric because 1 ≠ 2. For a relation to be antisymmetric, it cannot contain such pairs where the elements are distinct.
- Rule of Reflexivity:
Antisymmetry does not require reflexivity, but if a relation is reflexive and antisymmetric, then it must be a partial order.
- Example: The “less than or equal to” (≤) relation on integers is both reflexive and antisymmetric, thus making it a partial order.
- Rule of Transitivity:
Antisymmetry does not imply transitivity. However, if a relation is antisymmetric and transitive, then it could be a partial order if it is also reflexive.
- Example: In the set {1,2,3}, consider the relation R = {(1,2),(2,3),(1,3)}. This relation is antisymmetric (no two distinct elements have bidirectional pairs) and transitive. If it were reflexive, it would be a partial order.
- Rule of Directed Graph Representation:
In the directed graph of an antisymmetric relation, there should be no two distinct nodes with both edges a→b and b→a.
- Example: For the relation R={(1,2),(2,3)} on {1,2,3}, the directed graph shows no bidirectional edges between distinct nodes, which satisfies antisymmetry.
- Rule of Checking Non-Existence of Pairs:
To check if a relation is antisymmetric, examine whether any pairs (a,b) and (b,a) exist in the relation where a ≠ b. If such pairs exist, the relation is not antisymmetric.
- Example: For the relation R = {(2,1),(1,2),(3,3)} on {1,2,3}, the presence of both (2,1) and (1,2) with 2 ≠ 1 indicates that the relation is not antisymmetric.
- Rule of Combining with Other Properties:
While antisymmetry alone is a distinct property, combining it with reflexivity and transitivity results in a partial order.
- Example: The subset relation (⊆) on a set of subsets is antisymmetric, reflexive (every set is a subset of itself), and transitive (if A⊆B and B⊆C, then A⊆C), making it a partial order.
Examples of Antisymmetric Relation
Here are several examples of antisymmetric relations, covering a range of contexts to illustrate the concept:
1. Less Than or Equal To Relation (≤)
- Set: Z (set of all integers)
- Relation: ≤ (less than or equal to)
- Explanation: The relation a ≤ b is antisymmetric because if a ≤ b, then a must be equal to b. For example, if 3 ≤ 5 and 5 ≤ 3, then 3 must equal 5, which is not possible.
2. Subset Relation (⊆)
- Set: The power set of {1,2 (i.e., {∅,{1},{2},{1,2}}
- Relation: ⊆ (subset of)
- Explanation: The relation A⊆B is antisymmetric because if A⊆B and B⊆A, then A must be equal to B. For example, if {1}⊆{1,2} and {1,2}⊆{1}, then {1} must equal {1,2}, which is not possible.
3. Equal to Relation (=)
- Set: Any set, e.g., {a,b,c}
- Relation: = (equality)
- Explanation: The relation a = b is antisymmetric because if a = b and b = a, then a must be equal to b, which is always true.
4. Divisibility Relation
- Set: N (set of natural numbers)
- Relation: / (divides)
- Explanation: The relation a/b (a divides b) is antisymmetric because if a/b and b/a, then aaa must be equal to b. For example, if 2/6 and 6/2, then 2 must equal 6, which is not true.
5. Matrix Representation
- Set: The set of all 2×2 matrices
- Relation: A ≤ B if A is less than or equal to B in terms of matrix entry-wise comparison.
- Explanation: If A ≤ B and B ≤ A, then A must be equal to B, satisfying antisymmetry in matrix comparisons.
6. Order on a Set
- Set: {1,2,3}
- Relation: R = {(1,2),(2,2),(3,3),(1,3)}
- Explanation: This relation is antisymmetric because there are no pairs (a,b) and (b,a) where a ≠ b.
7. Parent-Child Relationship (in a Family Tree)
- Set: The set of people in a family
- Relation: Parent of
- Explanation: If A is a parent of B and B is a parent of A, then A and B must be the same person, which is impossible in a family tree, hence the relation is antisymmetric.
Also Read: Reciprocal: Definition, Meaning and Solved Examples
FAQs
An antisymmetric relation is a binary relation where if (a, b) and (b, a) are both in the relation, then a must equal b. In other words, if two different elements are related in both directions, then they must be the same element.
Example of an asymmetric relation: “Is less than” between numbers. If 3 is less than 5, then 5 cannot be less than 3.
An antisymmetric function is a function where if f(x) = f(y), then x = y. For example, the function f(x) = x3 is antisymmetric because if x3 = y3, then x = y.
Symmetric relations mean that if A is related to B, then B is also related to A. Antisymmetric relations mean that if A is related to B and B is related to A, then A must equal B.
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