Antisymmetric Relation

In mathematics, relations are used to describe the relationship between the elements of two sets. They aid in the mapping of elements from one set (the domain) to elements from another set (the range), resulting in ordered pairs of the type (input, output). Functions are also particular forms of relations that can be employed to construct a correspondence between two quantities. A function can also be considered a subset of a relation. Mathematical relations serve to build a connection between any two items or things. A relation is a term used to explain the relationship between two items that are usually represented as an ordered pair (input, output) or as two separate objects (x, y). x and y are set elements in this case. Relations are used in a variety of fields, including computer science to construct relational database management systems (RDBMS). 

Relations are a subset of the cartesian product of two sets in mathematics. Assume that X and Y represent two sets. Let x ∈X (x is an element of set X) and y ∈Y be two variables. The cartesian product of X and Y is thus given as the collection of all possible ordered pairs, denoted by  (x, y). To put it another way, a relation states that each input will result in one or more outputs.

Relation Types

Different types of connections might exist between two sets, necessitating the use of different types of relations to characterize these connections. The following are the most common forms of relations:

Empty Relation

An empty relation is one in which no element of a set is mapped to another set’s element or to itself. it is denoted by R =  ∅

For instance,

  P = {3, 7, 9} and the relation on P, R = {(x, y), where x + y = 76} . Because no two elements of P sum up to 76, this will be an empty relation..

Universal Relation

A universal relation is one in which all of the components of one set are mapped to all of the elements of another set or to themselves. R = X x Y denotes that each element of X is connected to every element of Y. For instance, P ={ 3, 7, 9}, Q = {12, 18, 20}, and R ={ (x, y), where x< y}.

Identity Relation

When all of the members in a set are related to each other, the relation is called an identity relation. For any I = {(x, x) : for all x ∈ X}. For instance, if P = {3, 7, 9} then I ={ (3, 3), (7, 7), (9, 9)}

Inverse Relation

An inverse connection exists when the members of one set are the inverse pairings of the elements of another set. In other words, a relation’s inverse is also a relation. 

R-1 denotes the inverse of a relation R. 

R-1 = {(y, x): (x, y)∈ R}

Reflexive Relation

A reflexive relation exists when all of the items in a set are mapped to themselves.  

As a result, if x  ∈ X, a reflexive relation is defined as (x, x) ∈ R.

  P ={ 7, 1} is a reflexive relation, as is R ={ (7, 7), (1, 1)}.

Symmetric Relation

If one set, X, contains ordered pairings (x, y), as well as the reverse of these pairs, the relation is said to be symmetric (y, x). In other words, for the relation to be symmetric, if (x, y) ∈ R, then (y, x) R. If P = {3, 4}, then R ={ (3, 4), (4, 3)} can be used as a symmetric relation.

Transitive relation

If (x, y ) ∈ R and (y, z) ∈ R are true, then R is a transitive relation if (x, z)  ∈R is true. For example, if P = {p, q, r}, then R = {(p, q), (q, r), (p, r)} is a transitive connection.

Equivalence relation

A symmetric, transitive, and reflexive relation combinedly establish an equivalence type of relation.

Let’s look at what antisymmetric relations mean presently.

If there are no pairs of unique elements of A that are related to each other by R, the relation R is said to be antisymmetric.

It is denoted mathematically as:

For every a, b ∈ A

If (a,b)∈ R and (b,a)∈ R are true, then a=b.

Equivalently, For every a, b ∈ A

If (a,b)∈ R and a≠ b are true, then (b,a)∈ R cannot be true.

The relation R on the set X is as follows:

Because x is divisible by y, an element x in X is connected to an element y in X.

That is, ( x, y ) ∈ R if and only if y divides x. Now, we’ll see if R is an antisymmetric relation.

Assume that ( x, y ) ∈ R and ( y, x )∈ R are true.

This means that x is divisible by y and that y is divisible by x.

Only if x = y is this possible.

Let’s look at some specific x and y values to see if this is true.

Assume x = 4 and y = 2.

We can say (4,2) ∈ R but (2,4) ∉ R since 4 is divisible by 2, but 2 is not.

As a result, R is an antisymmetric relation.

Points to Remember

• A collection of ordered pairs is referred to as a relation.
• If there are no pairs of unique elements of A that are related to each other by R, the relation R is said to be antisymmetric.