Definition Properties and Examples of 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 in Discrete Mathematics
1. What is an asymmetric relation in mathematics?
An asymmetric relation is a relation where if (a, b) belongs to the relation, then (b, a) does not belong to the relation. In set theory, a relation R on a set A is asymmetric if:
If (a, b) ∈ R, then (b, a) ∉ R for all a, b ∈ A.
- This automatically means no element is related to itself.
- So, if R is asymmetric, then (a, a) ∉ R for all a.
- Example: The relation “less than” (<) on real numbers is asymmetric.
2. Can you give an example of an asymmetric relation?
A common example of an asymmetric relation is the “less than” relation (<) on numbers. For real numbers:
- If 3 < 5, then 5 < 3 is false.
- Thus, if (3, 5) ∈ R, then (5, 3) ∉ R.
- Also, no number is less than itself, so (a, a) ∉ R.
3. What is the difference between asymmetric and antisymmetric relations?
The key difference is that an asymmetric relation never allows both (a, b) and (b, a), while an antisymmetric relation allows them only if a = b.
- Asymmetric: If (a, b) ∈ R, then (b, a) ∉ R, and (a, a) is never allowed.
- Antisymmetric: If (a, b) and (b, a) ∈ R, then a = b.
- Every asymmetric relation is antisymmetric, but not every antisymmetric relation is asymmetric.
4. Is every asymmetric relation also antisymmetric?
Yes, every asymmetric relation is also antisymmetric. If a relation is asymmetric, then:
- Whenever (a, b) ∈ R, (b, a) ∉ R.
- So it is impossible for both (a, b) and (b, a) to be in R unless a ≠ b never occurs.
- This satisfies the definition of antisymmetric automatically.
5. Can an asymmetric relation be reflexive?
No, an asymmetric relation cannot be reflexive because reflexivity requires (a, a) ∈ R for all a. In contrast:
- Asymmetry implies if (a, a) ∈ R, then (a, a) ∉ R.
- This creates a contradiction.
- Therefore, an asymmetric relation is always irreflexive.
6. How do you check if a relation is asymmetric?
To check if a relation is asymmetric, verify that no pair and its reverse both appear in the relation. Follow these steps:
- Step 1: List all ordered pairs in the relation.
- Step 2: For each (a, b), check whether (b, a) also appears.
- Step 3: Ensure no element relates to itself (no (a, a)).
7. What properties does an asymmetric relation have?
An asymmetric relation has specific logical properties that distinguish it from other types of relations.
- It is always irreflexive.
- It is automatically antisymmetric.
- It cannot be symmetric.
- It may or may not be transitive.
8. Is the “less than or equal to” relation asymmetric?
No, the “less than or equal to” (≤) relation is not asymmetric because it allows (a, a). For example:
- Since 3 ≤ 3, (3, 3) ∈ R.
- Asymmetry requires (a, a) ∉ R.
- Therefore, ≤ is not asymmetric.
9. Can an asymmetric relation be transitive?
Yes, an asymmetric relation can be transitive, but transitivity is not required. A relation R is transitive if:
If (a, b) ∈ R and (b, c) ∈ R, then (a, c) ∈ R.
- The “less than” (<) relation is both asymmetric and transitive.
- Some asymmetric relations may fail to satisfy transitivity.
10. Why is an asymmetric relation always irreflexive?
An asymmetric relation is always irreflexive because allowing (a, a) would violate asymmetry. By definition:
- If (a, a) ∈ R, asymmetry requires (a, a) ∉ R.
- This creates a contradiction.
- Therefore, for all a, (a, a) ∉ R.
