Antisymmetric Relation is a type of binary relation on a set where any two distinct elements related to each other in one direction cannot be related in the opposite direction. For example, consider the relation "less than or equal to" (≤) on the set of integers. This relation is antisymmetric because if a ≤ b and b ≤ a, then a must be equal to b. This article deals with Antisymmetric Relation including their definition, examples, as well as properties.
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What is Relation in Maths?
A relation refers to a set of ordered pairs, where each pair consists of elements from two sets. These sets can be the same or different. A relation R between two sets A and B is defined as a subset of the Cartesian product A × B. In other words, if (a, b) is an ordered pair in the relation R, it means that there is some kind of relationship between a and b.
For example, let's consider two sets:
- A = {1, 2, 3}
- B = {4, 5, 6}
A relation between A and B could be R={(1,4), (2,5), (3,6)}.
Types of Relation
There can be various types of relations in mathematics, i.e.,
- Reflexive Relation
- Symmetric Relation
- Transitive Relation
- Antisymmetric Relation
- Asymmetric Relation
- Equivalence Relation
In this article, we will learn about antisymmetric relation in detail.
What is Antisymmetric Relation?
An antisymmetric relation is a relation in which is two elements of set are related with relation R i.e., first element R second element and second element R first element then, first element is equal to second element.
In other words, antisymmetric relation is defined as if aRb and bRa then, a = b. A relation R = {(a, b) → R | a ≤ b} is an asymmetric relation since a ≤ b and b ≤ a implies a = b.
Antisymmetric Relation Definition
The relation is said to be an antisymmetric relation if in a set S the two elements p and q are related with relation R then, p = q. Also, if for every (p, q) ∈ R, (q, p) ∉ R then, R is antisymmetric. Mathematically, the antisymmetric relation is defined as:
If x and y are two elements in set X and R is a relation then, conditions for relation to be antisymmetric:
(xRy and yRx) ⇒ (x = y) ∀ x, y ∈ X
or
(x, y) ∈ R then, (y, x) ∉ R
Examples of Antisymmetric Relations
There are multiple examples of antisymmetric relation. Some of these examples are listed below.
- Less than operation on two elements.
- Equality relation on any set.
- Divisibility relation
- Subset
Let's consider an example to check for antisymmetry:
Example: If relation R = {(1, 1), (4, 7), (7, 4)} then, find the given relation is an antisymmetric relation or not?
Solution:
R = {(1, 1), (4, 7), (7, 4)}
The above relation is antisymmetric as
- (1, 1) ∈ R and (1, 1) ∈ R and 1 = 1.
- (4, 7) ∈ R and (7, 4) ∈ R and 4 ≠ 7.
R is not an antisymmetric relation.
Properties of Antisymmetric Relations
The properties of antisymmetric relations are listed below:
- Empty relation on any set is always antisymmetric.
- A relation can be symmetric and antisymmetric at same time.
- If R is an antisymmetric relation, then R-1 is also antisymmetric.
- It R1 and R2 are two antisymmetric relations, then R1 ∩ R2 is also antisymmetric.
- In the matrix representation of antisymmetric relation, either Mij = 0 or Mij ≠ 0 when i ≠ j.
Number of Antisymmetric Relations
The formula for number of antisymmetric relations with n-elements is given by:
Total number of antisymmetric relation = 2n × 3 [n(n-1)]/2
How to Check Relation is Antisymmetric or not?
To check whether the given relation is antisymmetric or not follow the below steps.
- First, check for every (a, b) in the given relation the existence of (b, a).
- If (b, a) is present and b ≠ a then, relation is not antisymmetric.
- If for every (a, b) there is (b, a) and in all the pairs a = b then, relation is antisymmetric.
- If (b, a) is absent then, relation is antisymmetric.
Symmetric and Antisymmetric Relations
Below table represents the difference between the symmetric and antisymmetric relation.
Characteristics | Symmetric Relation | Antisymmetric relation |
|---|---|---|
Definition | A relation R is symmetric when two elements p and q of set A if p is related to q then, q is also related to p. | A relation R is antisymmetric when two elements p and q of set A if p is related to q and q is related to p then p is equal to q or if (p, q) belongs to R then, (q, p) does not belongs to R. |
Mathematical Representation | pRq ⇔ qRp | (pRq and qRp) ⇒ (p = q) ∀ x, y ∈ X or (p, q) ∈ R then, (q, p) ∉ R |
Example | Parallel lines if a || b then b || a | Greater than if a > b then, b≯ a. |
Conclusion
From the above discussion we can conclude that a relation R is said to be an antisymmetric relation when if x and y holds the relation R i.e., if xRy and yRx then, x = y. The formula for calculating the total number of antisymmetric relations from a set of n elements is 2n × 3 [n(n-1)]/2. Also, we have learnt that a relation can be symmetric or antisymmetric at a same point of time.
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Sample Problems on Antisymmetric Relations
Example 1: Check whether the relation R = {(1,4), (2,5)} is antisymmetric or not?
Solution:
R = {(1,4), (2,5)}
The above relation is antisymmetric as
- (1, 4) ∈ R and (4, 1) ∉ R.
- (2, 5) ∈ R and (5, 2) ∉ R.
R is antisymmetric.
Example 2: Prove the given relation R = {(2,2), (3,7)} is an antisymmetric relation?
Solution:
R = {(2, 2), (3, 7)}
The above relation is antisymmetric as
- (2, 2) ∈ R and (2, 2) ∈ R and 2 = 2.
- (3, 7) ∈ R and (7, 3) ∉ R.
R is antisymmetric.
Example 3: Find the number of antisymmetric relations on set V with 2 elements.
Solution:
The total number of antisymmetric relation = 2n × 3 [n(n-1)]/2
The total number of antisymmetric relation on given set V = 22 × 3 [2(2-1)]/2
The total number of antisymmetric relations on given set V = 22 × 3
The total number of antisymmetric relations on given set V = 12