Anti-Symmetric

The set theory relies heavily on relationships and their types. You should be aware that sets, relations, and functions are interconnected. The antisymmetric connection is a construct based on symmetric and asymmetric relationships in discrete mathematics. Simply put, an antisymmetric definition of a set is one in which there is no ordered pair and its inverse in the relation.

Basics of Antisymmetric Relation

For a binary relation on a set A, a relation is an antisymmetric relation. There are no two different elements of A, respectively, which are related to the other through R. Aside from antisymmetric, there are also reflexive, irreflexive, symmetric, asymmetric, and transitive relations.

For all a and b in A, the relation R is antisymmetric; if R(x, y) holds with x y, then R(y, x) must not. If R(x, y) and R(y, x) are equal, then x = y. As a result, when (x,y) is related to R, (y, x) is not. x and y are the elements of set A in this case.

Rules of Antisymmetric Relation

Relationships can often follow any set of rules. For example, consider the relation ‘is divisible by,’ an ordered pair relation in the set of integers. An ordered pair (x, y) can be found for a relation R, where x and y are whole numbers or integers, and x is divisible by y. On the other hand, Antisymmetric relations do not have to hold R(x, x) for any value of x. It is a characteristic of reflexive relationships.

Asymmetric & Antisymmetric

There are various types based on the precise properties that a relation must satisfy when it comes to relations. Asymmetric and antisymmetric relations are two examples of these types of relationships. One of the two relations we’ve discussed so far is asymmetric, whereas the other is antisymmetric. Let’s look at each of these relationships and try to figure out which one is which.

• R is an asymmetric relation that has the following property:

If (x, y) is in R, then (y, x) is not R.

As a result, if an element x is related to an element y according to some rule, y cannot be related to x according to the same rule. To put it another way, in an unbalanced relationship, you can’t have it both ways.

T is an antisymmetric relation that has the following property:

• If (x, y) and (y, x) are in T, then x = y.

For example, if an element x is related to an element y, and the element y is also related to the element x, then the elements x and y must be the same. As a result, the only way an antisymmetric relationship can go both ways is if x = y.

Okay, similar names, but an asymmetric relation differs from an antisymmetric relation in that an asymmetric relation cannot go both ways. In contrast, an antisymmetric relation can go both ways if the two elements are equal.

Examples of Relations

After learning about what is antisymmetric, consider our two real-world instances of relations once more, and try to figure out which is asymmetric and which is antisymmetric. For example, consider the relation G, ordered pairs (f, s), with f being the father of s. For this relationship to be asymmetric, it must be the case that if (f, s) is in G, then (s, f) cannot be in G. This makes perfect sense! If f is the father of s, then s cannot possibly be f’s father. That’s impossible from a biological standpoint! As a result, G is asymmetric, and we know it isn’t antisymmetric because the relationship can’t be symmetric in both directions.

Example of Antisymmetric Relation

Question 2: A = 1, 2, 3, 4, and R is the relation on set A. On set A, find the antisymmetric relation.

Solution:

On the set A = 1, 2, 3, 4, the antisymmetric relation is:

(1, 1), (2, 2), (3, 3), (4, 4) are the values of R.

If there are no pairs of distinct elements, we can conclude that a binary relation on a set is antisymmetric.

Conclusion

For a binary R, a relation becomes an antisymmetric relation. It is the clear antisymmetric definition. There are no two different elements of A, in this case, each of which is related to the other through R. Aside from antisymmetric, there are also reflexive, irreflexive, symmetric, asymmetric, and transitive relations.