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Types of Relations

In mathematics, Relations are mappings between two sets, X and Y and they are of several types. They connect elements of one set to another. Relations and their types are a very important concept in Set Theory. Functions are special types of relations and are one of the major use of relation.

A very abstract representation of a relation between two sets, X and Y.
A very abstract representation of a relation between two sets, X and Y.
[Source: Wikimedia Commons]

Definition of Relation

In mathematical terms, a relation \(R\) is defined as a subset of the Cartesian Product \(A \times B\) of two sets \(A\) and \(B\). Note that \(A\) and \(B\) have to be non-empty for this. 

What is the Cartesian Product?

Consider a set \(A = \{1, 3, 7\}\) and a set \(B = \{2, 5 \}\). Then the Cartesian Product \(A \times B\) is given by,

\(A \times B = \{ (1,2), (1,5), (3,2), (3,5), (7,2), (7,5) \}\)

In other words, it is the set of all possible ordered pairs, whose first element is from \(A\) and whose second element is from \(B\). The first set \(A\) is called the domain and the second set \(B\) is called the co-domain.

Types of Relations in Mathematics

Relations are of many types. Here, we list out and then discuss the prominent ones. 

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

Here are some details about the various types of relations.

Empty Relation

  • These types of relations have no elements at all, and is an empty set.
  • It means that there is no association or mapping of an element from one set A to the other set B. 
  • It can be represented as \(R = \phi\).

Universal Relation

  • It is the opposite of the empty relation.
  • This relation is defined from and to the same set A.
  • All elements in A are related to all elements in A.
  • In short, it can be represented as \(R = A \times B\).

Identity Relation

  • It is the relation where every element of A is related to itself, and itself only.
  • For example, when \(A = B = \mathbb{R}\), the set of real numbers, the relation \(x=y\) is an identity relation.

Symmetric Relation

  • A relation \(R\) is symmetric, if \((x, y) \in R\) implies \((y, x) \in R\), for all \(x \in A\) and \(y \in B\).
  • For example, the relation “x is the additive inverse of y” is symmetric.

Reflexive Relation

  • A relation is reflexive if every element of the first set \(A\) is related to itself.
  • Mathematically, we write, \(R\) is reflexive, if for \(x \in A\), \((x,x) \in R\).

Transitive Relation

  • A relation \(R\) is transitive, if when \((x,y) \in R\) and \((y,z) \in R\), then \((x,z) \in R\), where \(x, y, z \in A\).
  • An example is the relation “x is greater than y”.

Inverse Relation

  • A relation \(S\) is an inverse relation of a relation \(R\), if \((y,x) \in S\) for all \((x,y) \in R\). 
  • This relation \(S\) is denoted by \(R^{-1}\). 

Equivalence Relation

  • A relation is said to be an equivalence relation if it is reflexive, symmetric, and transitive.
  • For example, the relation “is parallel to” is an equivalence relation on a set \(A\) of all lines in a plane. This is because:
    Reflexivity: Every line is parallel to itself.
    Symmetry: If line \(l\) is parallel to line \(m\), then \(m\) is also parallel to \(l\).
    Transitivity: If \(l\) is parallel to \(m\) and \(m\) is parallel to \(n\), then \(l\) must be parallel to \(n\).
The three conditions for an equivalence relations.
The three conditions for an equivalence relation. (Source)

Applications of Relations

Relations are an important concept in set theory and its operations. Therefore, they play an important role in other concepts like functional analysis. The applications are broad-ranging and sets the foundations for many other fields in set theory.

Solved Examples

Question 1. Find the Cartesian Product of \(A= \{1, 2, 3 \}\) and \(B= \{ x, y, z \}\).

Solution. We have, \(A \times B = \{ (1,x), (1,y), (1,z), (2,x), (2,y), (2,z), (3,x), (3,y), (3,z) \}\).

All relations from \(A\) to \(B\) have to be subsets of this set.

Question 2. Show that “is congruent to” is an equivalence relation on triangles in a plane.

Solution. Let us consider the set all triangles in a plane. Now we check if the relation “is congruent to” satisfies reflexivity, symmetry, and transitivity.

  1. Reflexivity: Every triangle is congruent to itself, so the relation is reflexive.
  2. Symmetry: If \(\triangle A\) is congruent to \(\triangle B\), then \(\triangle B\) is congruent to \(\triangle A\). Thus, the relation is symmetric.
  3. Transitivity: If \(\triangle A\) is congruent to \(\triangle B\) and \(\triangle B\) is congruent to \(\triangle C\), then we have \(\triangle A\) is congruent to \(\triangle C\). Thus the relation is transitive.

From the observations, we can see that “is congruent to” is an equivalence relation on the set of triangles in a plane.

FAQs

What is a relation in mathematics?

A relation is a mapping, or association, from one set A to another set B. Elements from one set are linked to elements of another set.

What are the types of relations in mathematics?

The major types of relations in mathematics are:
1. Empty relation
2. Universal relation
3. Identity Relation
4. Symmetric Relation
5. Reflexive Relation
6. Transitive Relation
7. Inverse Relation
8. Equivalence Relation

What is an equivalence relation?

A relation is an equivalence relation if it satisfies conditions for reflexive, symmetric and transitive relation.

What is the difference between a relation and a function?

A relation is any kind of mapping between two sets.
A function is a special kind of relation. Each element in the first set has one and only one mapping in the second set.

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