Relations and Functions

Types of Relations

If a person pointing towards a boy says he is the son of my wife. What is the relationship between the boy and the man? No doubt, a relation of father-son. By relation, we understand a connection or a link between the two. In general term, we understand and can find out the relationship among the two. In the set theory, a relation is a way of showing a connection or relationship between any two sets.

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A connection between the elements of two or more sets is Relation. The sets must be non-empty. A subset of the Cartesian product also forms a relation R. A relation may be represented either by Roster method or by Set-builder method.

Let A and B be two sets such that A = {2, 5, 7, 8, 10, 13} and B = {1, 2, 3, 4, 5}. Then,

R = {(x, y): x = 4y – 3, x ∈ A and y ∈ B} (Set-builder form)

R = {(5, 2), (10, 3), (13, 4)} (Roster form)


Types of Relations or Relationship

Let us study about the various types of relations.


Empty Relation

If no element of set X is related or mapped to any element of X, then the relation R in A is an empty relation, i.e, R = Φ. Think of an example of set A consisting of only 100 hens in a poultry farm. Is there any possibility of finding a relation R of getting any elephant in the farm? No! R is a void or empty relation since there are only 100 hens and no elephant.

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Universal Relation

A relation R in a set, say A is a universal relation if each element of A is related to every element of A, i.e., R = A × A. Also called Full relation. Suppose A is a set of all natural numbers and B is a set of all whole numbers. The relation between A and B is universal as every element of A is in set B. Empty relation and Universal relation are sometimes called trivial relation.

Identity Relation

In Identity relation, every element of set A is related to itself only. I = {(a, a), ∈ A}. For example, If we throw two dice, we get 36 possible outcomes, (1, 1), (1, 2), … , (6, 6). If we define a relation as R: {(1, 1), (2, 2), (3, 3), (4, 4), (5, 5), (6, 6)}, it is an identity relation.

Inverse Relation

Let R be a relation from set A to set B i.e., R ∈ A × B. The relation R-1 is said to be an Inverse relation if R-1 from set B to A is denoted by R-1 = {(b, a): (a, b) ∈ R}. Considering the case of throwing of two dice if R = {(1, 2), (2, 3)}, R-1 = {(2, 1), (3, 2)}. Here, the domain of R is the range of R-1 and vice-versa.

Reflexive Relation

If every element of set A maps to itself, the relation is Reflexive Relation. For every a ∈ A, (a, a) ∈ R.

Symmetric Relation

A relation R on a set A is said to be symmetric if (a, b) ∈ R then (b, a) ∈ R, for all a & b ∈ A.

Transitive Relation

A relation in a set A is transitive if, (a, b) ∈ R, (b, c) ∈ R, then (a, c) ∈ R, for all a, b, c ∈ A

Equivalence Relation

A relation is said to be equivalence if and only if it is Reflexive, Symmetric, and Transitive. For example, if we throw two dices A & B and note down all the possible outcome.

Define a relation R= {(a, b): a ∈ A, b ∈ B}, we find that {(1, 1), (2, 2), …, (6, 6) ∈ R} (reflexive). If {(a, b) = (1, 2) ∈ R} then, {(b, a) = (2, 1) ∈ R} (symmetry). ). If {(a, b) = (1, 2) ∈ R} and {(b, c) = (2, 3) ∈ R} then {(a, c) = (1, 3) ∈ R} (transitive)

Solved Example for You

Problem: Three friends A, B, and C live near each other at a distance of 5 km from one another. We define a relation R between the distances of their houses. Is R an equivalence relation?

Solution: For an equivalence Relation, R must be reflexive, symmetric and transitive.

  • R is not reflexive as A cannot be 5 km away to itself.
  • The relation, R is symmetric as the distance between A & B is 5 km which is the same as the distance between B & A.
  • R is transitive as the distance between A & B is 5 km, the distance between B & C is 5 km and the distance between A & C is also 5 km.

Therefore, this relation is not equivalent.

Q1. What are the types of relation in maths?

A1. There are 9 types of relations in maths namely: empty relation, full relation, reflexive relation, irreflexive relation, symmetric relation, anti-symmetric relation, transitive relation, equivalence relation, and asymmetric relation.

Q2. Are all functions relations?

A2. A function is a kind of interrelationship among objects. Moreover, a function defines a set of finite lists of objects, one for each combination of possible arguments. Besides, a relation is another kind of interrelationship among object in the world of discourse. Furthermore, both function and relation are defined as a set of lists. And every function is a relation but not every relation is a function.

Q3. What is a void relation?

A3. We can define void relation as a relation R in a set A, where no element of set A is related to any element of A. So, R = ɸ which is a subset of A × A.

Q4. What is a universal relation?

A4.We can define universal relation as a set A when A × A ⊆ A × A. In simple words, it is a relation if each element set A is related to every element of A. Sometimes, we refer to it as trivial relations.

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2 responses to “Relations”

  1. Al quba says:

    The example for onto function doesn’t qualify as a function in the first place. Does it??
    It is a relation but not a function because a single element in the domain has been mapped to two elements in the co domain. Isn’t it??
    Please tell me if I’m correct or not.
    It is really confusing.

    • Eli Simeon says:

      An onto function exists if and only the co-domain is equal to the range that is every element in set A (the domain) is mapped to every element in set B (the range/codomain) i.e without leaving out any element. Irrespective of whether it is a one to one mapping or not. Therefore it is a function. Put simply, take set A as a set of sons and set B as a set of fathers, a function requires that every son has one father (which is normal) yet every father can have more than one son(which is also normal) so if one element in set A maps to more than one element in set B it is not a function (and we will need DNA test to know who really is the father) but if more than one element in set A maps to one element in set B it is still a function (the elements are just brothers). What and onto function requires is that every father has a son. Sorry if I made it a bit complicated I feel that if I continue I might make it worse just it study a bit more and from different sources, videos or books and you will understand it better.

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