Types Of Relations

The term relations is the most important part of set theory in mathematics. It is essential for basic knowledge of calculus. In this article, we will know the actual concept of the relations and other aspects and their types. With this, students will know the very concept of the related chapter. Further, you can solve the problems regarding it. By learning the types, they can define the types of it. They can clear their theories about the ut understanding the fundamentals.  

Relation

A relation is a connection between sets of values. 

Given sets X and Y, the Cartesian product X × Y is defined as {(x, y) | x ∈ X and y ∈ Y}, and its elements are called ordered pairs.

Types of relations 

The relation shows the following types of relations in the set theory of calculus. 

Identity Relation

Identity relation is the one in which every element maps to itself only. In other words, we can say, Every element is related to itself only. A= {(a, a):  ∈ A} on A is called the identity relation on A

Universal  relation

A relation is a type of relation that shows universality if every component of any given set is mapped to all the components of another set or the set itself. In other words, a relation R in a set A is termed a universal relation if each element of A is related to every element of A, 

i.e., R = A × A.

Symmetric Relation

Symmetric relation is a type of binary relation. An example is the relation “is equal to” because if a = b is true, then b = a is also true. Formally, a binary relation R over a set X is symmetric if:

∀a, b = X(aRb ⇒ bRa). 

Antisymmetric Relation

A binary relation on X is any subset R of X X X. Given a, b E X, write an Rb if and only if (a, b) ER, which means that an Rb is a shorthand for (a, b) E R. 

This can be written in the notation of first-order logic as

a, b) ∈ R and (b, a) ∈ R 

⇒ a = b ∀ a, b ∈ A

Transitive Relation

A relation R on a set X is transitive if, for all elements a, b, c in X, whenever R relates a to b and b to c, then R also relates a to c. Each partial order, as well as each equivalence relation, needs to be transitive.

It is denoted as 

(a, b) ∈ R and (b, c) ∈ R ⇒ (a, c) ∈ R, ∀ a, b, c ∈ A

Equivalence Relation

the relation R is reflexive, then all the elements of the given set are mapped with itself, such that for all a∈Q, then (a, a)∈R.

The relation R is said to be symmetric on set Q, if (a,b)∈R, then (b, a)∈R, such that a,b∈Q.

Reflexive Relation

Another type of relation shows that homogeneous binary relation R on a set X is reflexive if it relates every element of X to itself.

It shows that each element of set B is mapped with itself, such that an∈A,(a, a)∈R.

Empty relation 

The relation is the very first type of relationship that shows the relationship between sets of values. In maths, the relation is between the x-values and y-values of ordered pairs. The set of all x-values is called the domain, and the set of all y-values is called the range.

The empty set is referred to as the “null set”.

 However, the null set is a distinct notion within the context of measure theory, in which it describes a set of measure zero (which is not necessarily empty). The empty set may also be called the void set.

Condition: 

No element of set P is mapped with another set Q or set P itself. R= shows the empty relation ∅.

Examples

Suppose there are 200 carrots in the fruit bucket. There’s no chance of finding a relation R of getting any sweet potato in the basket. So, R is Void as it has 200 carrots and no sweet potato. 

The empty set has the following properties:

Its only subset is the empty set itself:

VA: ACO⇒ A=Ø

The empty set is a subset of A:

VA: Ø CA

The union of A with the empty set is A: VA: AUØ = = A

The intersection of A with the empty set is

the empty set:

VA: ANØ = 0 –

The Cartesian product of A and the empty set is the empty set: VA: AxØ = 0

Relation in maths 

In mathematics, the relation is between the x-values and y-values of ordered pairs. The set of all x-values is called the domain, and the set of all y-values is called the range.

Relations arise in many scientific disciplines, as well as in many branches of mathematics and logic; there is considerable variation in terminology. Aside from the set-theoretic extension of a relational concept or term, the term relation can also be used to refer to the related logical entity, either the systematic understanding, which is the entirety of intensions or abstract properties transferred by all aspects in the relation, or else the symbols denoting these elements and intentions.