Antisymmetric Relation is a relation R of a set A is antisymmetric if (a,b) ∈ R and (b,a) ∈ R, then a=b. Relations defined between sets and their types are an essential aspect of set theory. Sets indicate the collection of ordered elements, while relations and functions are there to denote the operations performed on elements in the sets. A relation between two sets A and B is a collection of ordered pairs of elements from A and B that satisfy the relation condition. It is a subset of the cartesian product AXB, of the sets A and B

There are different types of relations that can connect elements in two sets or within the same set.

1. Empty relation
2. Identity relation
3. Universal relation
4. Symmetric relation
5. Anti-Symmetric Relation
6. Transitive relation
7. Equivalence relation
8. Inverse relation
9. Reflexive relation

Antisymmetric Relation Definition

An antisymmetric relation is a type of relation in mathematics that follows a special rule.

Let’s say we have a set A, and a relation R is defined on it. The relation R is antisymmetric if the following condition is true:

If a is related to b and b is related to a, then a must be equal to b.

In symbols, this means:

If (a, b) is in R and (b, a) is also in R, then a = b
Where a and b are elements from the set A.

Example of Antisymmetric Relation: If we have a set R, where ‘a is less than equal to b’ where both a and b belong to R, then we see that for any a is less than equal to b as well as for the reverse to be true we must have a=b. Suppose a=4, b=5, then ‘a is less than equal to b’ is satisfied but ‘b less than equal to a’ is not satisfied. Thus this relation is an Antisymmetric Relation.

If we are given a set A and a relation on this set A, then the condition that must be satisfied for becoming an antisymmetric relation is given below:

\( aRb\ \&\ bRa\Rightarrow a=b \), where \( a,\ b\in A \)

We can also write the condition as,

\( \left(a,\ b\right)\in R\ \&\ \left(b,\ a\right)\in R\Rightarrow a=b,\ \forall\ a,\ b\in A \)

OR

\( \left(a,\ b\right)\in R\ \&\ a\ne b\Rightarrow\left(b,\ a\right)\notin R,\ \forall\ a,\ b\in A \)

R will not be antisymmetric if \( \left(a,\ b\right)\in R\ \&\ \left(b,\ a\right)\in R\ \) but\( a\ne b \), for any \( a,\ b\in A \)

Graph of Antisymmetric Relation

Let us understand the characteristics of antisymmetric relations through a digraph. The relations between the elements of two sets or the same set can be represented with the help of digraphs that use vertices to represent elements and directional edges between vertices to represent a relation between elements. Consider the following digraph of some relation R defined on the elements of a set.

We have ordered pairs such as \( R=\left\{\left(a,\ c\right),\left(a,\ b\right),\left(c,\ b\right)\right\} \). We can say that \( \left(a,\ b\right)\in R\ \&\ a\ne b\Rightarrow\left(b,\ a\right)\notin R,\ \forall\ a,\ b\in A \). Thus this is an Antisymmetric Relation.

Here from the above digraph, we get the relation set as \( R=\left\{\left(a,\ a\right),\left(b,\ b\right),\left(c,\ c\right),\left(d,\ d\right),\left(a,\ d\right),\left(a,\ b\right),\left(c,\ a\right),\left(c,\ b\right),\left(b,\ d\right)\right\} \) for the mother set of \( A=\left\{a,\ b,\ c,\ d\right\} \)

We see that for this relation set satisfies both the condition of antisymmetry, such as the first condition,

\( \left(a,\ b\right)\in R\ \&\ \left(b,\ a\right)\in R\Rightarrow a=b,\ \forall\ a,\ b\in A \).

This condition is satisfied by the ordered pairs \( \left(a,\ a\right),\left(b,\ b\right),\left(c,\ c\right),\left(d,\ d\right) \), and also the second condition is satisfied such as,

\( \left(a.\ b\right)\in R\ \&\ a\ne b\Rightarrow\left(b,\ a\right)\notin R,\ \forall\ a,\ b\in A \)

This condition is satisfied by the ordered pairs \( \left(a,\ d\right),\left(b,\ d\right),\left(c,\ b\right),\left(c,\ a\right),\left(a,\ b\right) \)

Thus the above-given digraph is an Antisymmetric relation.

Now this above-given digraph is not an Antisymmetric relation.

For the mother set \( A=\left\{1,\ 2,\ 3,\ 4\right\} \), we get the relation set as \( R=\left\{\left(2,\ 2\right),\left(3,\ 3\right),\left(1,\ 4\right),\left(4,\ 1\right),\left(1,\ 3\right),\left(3,\ 1\right),\left(2,\ 1\right),\left(2,\ 3\right),\left(4,\ 3\right)\right\} \)

This relation satisfies neither of the conditions of the Antisymmetric matrix.

Here \( \left(a,\ b\right)\in R\ \&\ \left(b,\ a\right)\in R\ but\ a\ne b\ \), as have only (2, 2) and (3, 3), and not (1, 1) and (4, 4).

Also, we have (1, 4) and (4, 1) but \( 1\ne4 \), so the above digraph is an Antisymmetric relation.

Basically, we can conclude that if we are given a digraph where we have a bidirectional arrow between two different elements then it can be said that it is not an Antisymmetric relation. As in the above digraph, we have a bidirectional arrow between 1 and 3 as well as between 1 and 4, so we can directly say it is not an Antisymmetric relation.

Rules of an Antisymmetric Relation

Let us consider a relation ‘is divisible by’ on an ordered pair relation on a set of all integers. We can find an ordered pair (x,y) in relation ‘R’ and here, ‘x’ and ‘y’ are whole numbers or integers. Also, x is divisible by y. 

The properties of an antisymmetric relation say:

• If (x,y) is in R, then (y,x) should not be in R.
• And, if (x,y) and (y,x) are in R, then x = y

How to Prove a Relation is Antisymmetric

By antisymmetric relation, we can say that if we have two sets, and one element of the first set is related to one element of the other set by some relation. Then, the element of the second set is related to the same element of the first set, following the same relation. Also, this can only be true when both the elements from the two different sets are equal.

Example of Antisymmetric Relation: Consider a set A = {1, 2, 3, 4} and let R be a relation on set A. Let us now find the antisymmetric relation on A.

Let (1,1), (2,2), (3,3), (4,4) be the ordered pairs in the relation R on set A.

As we can see that there are no pairs of distinct elements, so, the relation R on set A is antisymmetric.

Properties of Antisymmetric Relations 

1. Empty Relation is Always Antisymmetric
   If there are no pairs in the relation (called an empty relation), then it is always antisymmetric—because there’s nothing in it to break the rule!
2. A Relation Can Be Both Symmetric and Antisymmetric
   Yes, it’s possible! For example, if a relation only contains pairs like (a, a), then it is both symmetric (because (a, a) = (a, a)) and antisymmetric (because there are no different elements like (a, b) and (b, a) with a ≠ b).
3. Inverse of an Antisymmetric Relation is Also Antisymmetric
   If a relation R is antisymmetric, then its reverse (called R⁻¹) is also antisymmetric. That means if we flip all the ordered pairs in R, the antisymmetric property still holds.
4. Intersection of Two Antisymmetric Relations is Also Antisymmetric
   If you have two antisymmetric relations, R₁ and R₂, and take only the pairs that are common to both, the result (R₁ ∩ R₂) will also be antisymmetric.
5. Matrix Form of Antisymmetric Relation
   When you show a relation using a matrix, for any two different positions i and j (i ≠ j), either:
   • The entry in row i and column j is 0, or
   • If it’s not 0, then the entry in row j and column i must be 0
     This means both sides can’t be non-zero unless the row and column are the same (i = j).

How to Check if a Relation is Antisymmetric?

To find out whether a relation is antisymmetric, you can follow these easy steps:

1. Look at Each Pair (a, b) in the Relation
   Go through each ordered pair in the relation and check if the reverse pair (b, a) is also there.
2. Check If Both (a, b) and (b, a) Exist
   • If you find both (a, b) and (b, a) in the relation and a is not equal to b,
     then the relation is not antisymmetric.
3. If Both Pairs Exist but a = b
   • If (a, b) and (b, a) are both in the relation and a = b (like (3, 3)),
     then it's okay, and the relation can still be antisymmetric.
      
4. If the Reverse Pair (b, a) is Missing
   • If for any pair (a, b), the pair (b, a) is not in the relation,
     then the relation follows the antisymmetric rule.

Symmetric, Asymmetric and Antisymmetric Relation

Difference between Symmetric and Antisymmetric Relation

The difference between Symmetric Relation and Antisymmetric Relation is given below.

Antisymmetric Relation Examples

Example 1: Check whether the given relationship is Antisymmetric or not.
\( R=\left\{\left(a,\ a\right),\left(b,\ b\right),\left(c,\ c\right)\right\}\ on\ X=\left\{a,\ b,\ c\right\} \)

Solution: In the given relation R we see that all the ordered pairs have dissimilar elements. Therefore R satisfies the following condition, \( \left(a,\ b\right)\in R\ and\ \left(b,\ a\right)\in R \),

Thus we can say that a=b and hence the given relation is an Antisymmetric Relation.

Example 2: If \( S=\left\{1,\ 2,\ 3,\ 4,\ 5\right\} \), then find the Antisymmetric Relation for set S.

Solution: We have to find such a relation that satisfies the following condition for becoming Antisymmetric.

\( \left(a,\ b\right)\in R\ and\ \left(b,\ a\right)\in R\Rightarrow a=b \)

With this condition we get,

\( R=\left\{\left(1,\ 1\right),\left(2,\ 2\right),\left(3,\ 3\right),\left(4,\ 4\right)\right\} \)

Thus R is the resultant Antisymmetric Relation.

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