What is an Asymmetric Relation?
In discrete mathematics, the opposite of symmetric relation is asymmetric relation. In a set X, if one element is less than another element, agrees with the one relation, then the other element will not be less than the first one. Therefore, less than (>), greater than (<), and minus (-) are examples of asymmetric relations.
We can even say that the ordered pair of set X agrees with the condition of asymmetric only if the reverse of the ordered pair does not agree with the condition. This makes it identical from symmetric relation, where even the exact opposite of their orders are reversed, the condition is satisfied. There are 8 types of relations, these are :
Empty Relation
Universal Relation
Identity Relation
Inverse Relation
Reflexive Relation
Symmetric Relation
Transitive Relation
Equivalence Relation
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Domain and Range
If there are two relations A and B and the relation for A and B is R (a,b), then the domain is stated as the set { a | (a,b) ∈ R for some b in B} and range is stated as the set {b | (a,b) ∈ R for some an in A}.
Asymmetric Relation Definition
Asymmetric relation is the opposite of symmetric relation but not considered as equivalent to antisymmetric relation.
In Set theory, A relation R on a set A is known as asymmetric relation if no (b,a) ∈ R when (a,b) ∈ R or we can even say that relation R on set A is symmetric if only if (a,b) ∈ R⟹(b, a) ∉R.
For example: If R is a relation on set A= (18,9) then (9,18) ∈ R indicates 18>9 but (9,18) R, Since 9 is not greater than 18.
Note- Asymmetric relation is the opposite of symmetric relation but not considered as equivalent to antisymmetric relation.
The mathematical operators -,< and > are asymmetric examples whereas =, ≥, ≤, are considered as the twins of () and do not agree with the asymmetric condition.
Asymmetrical Relation Properties
Some basic asymmetrical relation properties are :
A relation is considered as asymmetric if it is both antisymmetric and irreflexive or else it is not.
Limitations and opposite of asymmetric relations are considered as asymmetric relations. For example- the inverse of less than is also an asymmetric relation.
Every asymmetric relation is not strictly in the partial order.
Subsequently, if a relation is of a strict partial order, then it will be considered as transitive and symmetric.
An asymmetric relation should not have the convex property. For example, the strict subset relation is regarded as asymmetric and neither of the assets such as {3,4} and {5,6} is a strict subset of others.
A transitive relation is considered asymmetric if it is irreflexive or else it is not. For example: if aRb and bRa, transitivity gives aRa contradicting ir-reflexivity.
Asymmetric Relation Solved Examples
1. If X= (3,4) and Relation R on set X is (3,4), then Prove that the Relation is Asymmetric.
Solution: Give X= {3,4} and {3,4}∈ R
Clearly, we can see that 3 is less than 4 but 4 is not less than 3, hence
{3,4} ∈ R ⇒ {4,3}∉ R
Hence, it is proved that the relation on set X is symmetric
Conclusion
This is all about the definition and explanation of asymmetric relation and its different forms. Focus on how the concept has been explained. Understand the concept well so that you can answer questions judiciously.
FAQs on Asymmetric Relation
1. What is an asymmetric relation in set theory?
An asymmetric relation is a binary relation on a set A where if an element 'a' is related to an element 'b', then 'b' is never related to 'a'. Formally, a relation R is asymmetric if for all a, b in A, if (a, b) ∈ R, then (b, a) ∉ R. This means no ordered pair and its reverse can both exist in the relation.
2. What is a simple example of an asymmetric relation?
A classic example of an asymmetric relation is the 'greater than' (>) relation on a set of integers. For instance, in the set {1, 2, 3}:
- 3 > 2 is true, so (3, 2) is in the relation.
- However, 2 > 3 is false, so (2, 3) is not in the relation.
Since this holds for any pair of numbers, the 'greater than' relation is asymmetric. Another real-world example is the relation 'is the parent of'.
3. How does an asymmetric relation differ from a symmetric relation?
The key difference lies in the requirement for the reverse ordered pair:
- A relation is symmetric if (a, b) being in the relation implies that (b, a) must also be in the relation. Example: 'is equal to'.
- A relation is asymmetric if (a, b) being in the relation implies that (b, a) must not be in the relation. Example: 'is less than'.
Essentially, they are opposites. A non-empty relation cannot be both symmetric and asymmetric.
4. What is the crucial difference between an asymmetric and an antisymmetric relation?
This is a common point of confusion. The main difference involves how they treat pairs of the form (a, a).
- Asymmetric Relation: Forbids both (a, b) and (b, a) from coexisting. This rule is so strict that it also implies no element can be related to itself, meaning (a, a) can never be in an asymmetric relation.
- Antisymmetric Relation: States that if (a, b) and (b, a) both exist in the relation, it can only be because a = b. This allows for elements to be related to themselves, such as in the 'less than or equal to' (≤) relation, where (5, 5) is valid.
In short, an asymmetric relation is always irreflexive, while an antisymmetric relation is not necessarily so.
5. Is every asymmetric relation also antisymmetric? Explain why.
Yes, every asymmetric relation is also antisymmetric. The condition for a relation R to be antisymmetric is: if (a, b) ∈ R and (b, a) ∈ R, then a = b. In an asymmetric relation, the premise of this condition—that both (a, b) and (b, a) are in R—is never met. Because the 'if' part of the statement is always false, the condition is considered to be vacuously true. However, the reverse is not true; an antisymmetric relation like '≤' is not asymmetric because it allows pairs like (4, 4).
6. What is the connection between asymmetric, antisymmetric, and irreflexive relations?
The connection is a fundamental property in relation theory. A relation R is defined as asymmetric if and only if it is both antisymmetric and irreflexive.
- Antisymmetric: If (a,b) ∈ R and (b,a) ∈ R, then a=b.
- Irreflexive: For every a in the set, (a,a) ∉ R.
An asymmetric relation must satisfy both these conditions simultaneously. It cannot have any loops (irreflexive) and cannot have any two-way paths between different elements (antisymmetric).
7. How can you find the total number of possible asymmetric relations on a set with 'n' elements?
To find the number of asymmetric relations on a set with 'n' elements, consider the n² possible ordered pairs. We analyse pairs of distinct elements (a, b) and pairs of identical elements (a, a).
- For any pair of distinct elements {a, b}, there are three possibilities for a relation R to be asymmetric: (a, b) is in R, (b, a) is in R, or neither is in R. There are n(n-1)/2 such pairs of distinct elements.
- For any pair (a, a), it cannot be in an asymmetric relation (due to the irreflexive property).
Therefore, the total number of asymmetric relations is calculated as 3n(n-1)/2.
