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. InÂ the set theory, a relation is a way of showing a connection or relationship between any two sets.

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## Relation

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 âˆˆÂ A and y âˆˆÂ 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 = Î¦. 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.

**Browse more Topics under Relations And Functions**

- Relations
- Functions
- Types of Functions
- Representation of Functions
- Composition of Functions and Invertible Function
- Algebra of Real Functions
- Cartesian Product of Sets
- Binary Operations

### 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 Ã— 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), âˆˆ A}. For example, If we throw two dice, we get 36 possible outcomes, (1, 1), (1, 2), â€¦ , (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 âˆˆÂ A Ã— 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) âˆˆ 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 ofÂ R^{-1Â }and vice-versa.

### Reflexive Relation

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

### Symmetric Relation

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

### Transitive Relation

A relation in a set A is transitive if, (a, b) âˆˆ**Â **R, (b, c) âˆˆ R, then (a, c) âˆˆ R, for all a, b, c âˆˆ 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 âˆˆ A, b âˆˆ B}, we find that {(1, 1), (2, 2), â€¦, (6, 6) âˆˆ**Â **R} (reflexive). If {(a, b) = (1, 2) âˆˆ**Â **R} then, {(b, a) = (2, 1) âˆˆ R} (symmetry). ). If {(a, b) = (1, 2) âˆˆ**Â **R} and {(b, c) = (2, 3) âˆˆ**Â **R} then {(a, c) = (1, 3) âˆˆ**Â **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 = É¸ which is a subset of A Ã— A.

**Q4. What is a universal relation?**

**A4.**We can define universal relation as a set A when A Ã— A âŠ† A Ã— 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.

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.

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.