Asymmetric and Symmetric Relations
An asymmetric relation is a binary relation on a set where if (a,b) is in the relation, then (b,a) must not be in the relation for any elements a and b.
Example:
- The “is a parent of” relationship is asymmetric. If Alice is the parent of Bob, then Bob cannot be the parent of Alice.
A symmetric relation is a binary relation on a set where if (a, b) is in the relation, then (b,a) must also be in the relation for any elements a and b.
Example:
The “is a sibling of” relationship is symmetric. If Alice is a sibling of Bob, then Bob is also a sibling of Alice.
Difference Between Asymmetric and Symmetric Relations
|
Asymmetric Relation Vs Symmetric Relations |
||
|---|---|---|
| Property | Asymmetric Relations | Symmetric Relations |
| Definition | (a, b) in the relation implies (b, a) is not in the relation | (a, b) in the relation implies (b, a) is also in the relation |
| Direction of Relationship | One-way relationship | Two-way relationship |
| Example | “Is Less Than” (<<) | “Is Equal To” (==), “Is a Friend of” |
| Transitivity | Can be transitive or intransitive | Often transitive |
| Matrix Representation | Zeros on the main diagonal, no symmetric entries | Symmetric entries reflected across the main diagonal |
| Antisymmetry | Always antisymmetric. (Not the same as asymmetric) | May or may not be antisymmetric |
| Irreflexivity | Always irreflexive | May or may not be irreflexive |
Asymmetric Relation
A relation is a subset of the cartesian product of a set with another set. A relation contains ordered pairs of elements of the set it is defined on. A relation is a subset of the cartesian product of a set with another set. A relation contains ordered pairs of elements of the set it is defined on.
Table of Content
- What is Relation in Maths?
- What is Asymmetric Relations?
- Properties of Asymmetric Relations
- Asymmetric and Symmetric Relations
- Examples of Asymmetric Relations
- Conclusion: Asymmetric Relation
- Sample Problems on Asymmetric Relations
- FAQs On Asymmetric Relation
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