In simple words, a relation in set theory shows how elements from one set are connected to elements in another set. It helps us understand the link or pairing between items in different sets. Functions are a special kind of relation where each input has only one output. There are many types of relations, such as:

• Universal relation (everything is related),
• Identity relation (each element is related to itself),
• Empty relation (no elements are related),
• Reflexive, symmetric, transitive, anti-symmetric, and equivalence relations,
• And inverse relation (flips the direction of a relation).

Sets and Relations

In mathematics, sets and relations are closely linked. A relation helps us understand how two sets are connected.

If you have two sets, a relation tells you how the elements from one set are related to the elements of the other set.

For example, if none of the elements from one set are connected to the elements of the second set, it is called an empty relation.

A relation in math shows how values from two sets are connected.

Let’s say we have set x and set y and we write them as ordered pairs like (x, y).

• The first elements in each pair come from the first set – this is called the domain.
• The second elements come from the second set – this is called the range.

Example:

Given ordered pairs:
{ (1, 2), (-3, 4), (5, 6), (-7, 8), (9, 2) }

• Domain (first numbers in each pair): { -7, -3, 1, 5, 9 }
• Range (second numbers in each pair): { 2, 4, 6, 8 }

So, a relation helps us pair elements from one set with elements from another set.

What are Types of Relations?

There are 9 types of relations:

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

1. Empty Relation: When none of the components of set X is associated/mapped to any element of X, then the relation R in A is an empty relation, also referred to as the void relation, i.e R= ∅. This states that A relation R in a set A is termed to be an empty relation or void relation if none of the elements of A is associated with any component of A, i.e., R =∅ ⊂ A × A.

Example of empty relation: if there are 300 apples in the fruit bucket. There’s no possibility of finding a relation R of getting any banana in the basket. So, R is Void as it has 300 apples and no bananas.

2. Identity Relation: A relationship is supposed to be an identity relation if all the components of the given set are related to itself, generally denoted I or \(I_{A}\).

Condition: Let A be a non-empty set then the relation \(I_{A}\)= {(a, a): a ∈ A} on A is called the identity relation on A.

Example of Identity relation: If A = {1, 2, 3}, then the relation given by the set {(1, 1), (2, 2), (3, 3)} is called the identity relation on the given set A denoted by \(I_{A}\).

3. Universal Relation: In terms of theoretical definition, a relation is universal if every component of any given set is mapped to all the components of another set or the set itself. Universal relations are also called full relations as every element of a given set is linked to every element of the set itself.

Example of Universal relation: Suppose we have a set 1 that comprises all the natural numbers and another set 2 that consists of all whole numbers. Then we can say that the relation between 1 and 2 is universal as every element of set 1 is within set 2.

An empty plus universal relation is also called a trivial relation.

Condition: Consider R as a relation in a set, and A denotes a universal relation because, in this whole relation, each component of A is related to each component of A. i.e R = A × A.

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.

4. Symmetric Relation: A relation is supposed to be a symmetric one, in which the ordered pair of a given set plus the reverse ordered pair are present in the relation.

Condition: Let R signify a relation on a non-empty set A, then the relation R is said to be symmetric relation ⇔ (a, b) ∈ R ⇒ (b, a) ∈ R ∀ a, b ∈ A.

Example of Symmetric relation: If A = {3, 4}, then relation R = {(3, 3), (4, 4), (3, 4), (4, 3)} is symmetric ∵ if (a, b) ∈ R then (b, a) ∈ R ∀ a, b ∈ A.

5. Anti-Symmetric Relation: Let R be a relation on a non-empty set A, then the relation R is said to be anti-symmetric relation ⇔ (a, b) ∈ R and (b, a) ∈ R ⇒ a = b ∀ a, b ∈ A

Example of anti-symmetric relation: If A = {3, 4, 5}, then relation R = {(3, 3), (4, 4), (5, 5)} is anti-symmetric ∵ if (a, b) ∈ R and (b, a) ∈ R then a = b ∀ a, b ∈ A.

6. Transitive Relation: Let R signify a relation on a non-empty set A, then the relation R is said to be transitive relation ⇔ (a, b) ∈ R and (b, c) ∈ R ⇒ (a, c) ∈ R, ∀ a, b, c ∈ A.

Example of Transitive relation: If A = {3, 4, 5}, then relation R = {(3, 4), (4, 5), (3, 5)} is transitive ∵ if (a, b) ∈ R and (b, c) ∈ R then (a, c) ∈ R, ∀ a, b, c ∈ A.

7. Equivalence Relation: Let R be a relation on a non-empty set A, then the relation R is said to be the equivalence relation if R is reflexive, symmetric, and transitive.

Conditions: If 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.

The relation R on set Q, if (a,b)∈R and (b,c)∈R, then (a,c)∈R for all a,b,c ∈R is said to be a transitive relation.

Example of Equivalence relation: If A = {3, 4, 5}, then relation R = {(3, 3), (4, 4), (5, 5), (3, 5), (5, 3), (3, 4), (4, 5)} is an equivalence relation ∵ the relation R is reflexive, symmetric and transitive as shown above respectively.

8. Inverse Relation: If a set with elements holds the inverse pairs of another set, then the relation is termed inverse relation.

Condition: Let A and B be any two non-empty sets and R be the relation from A to B. Then, the inverse of R, denoted by \(R^{-1}\) is a relation from B to A and is given by; \(R^{-1}\) = {(b, a): (a, b) ∈ R}

Example of Inverse relation: If A = {a, c}, B = {d, e} and the relation R from A to B is given by: R = {(a, d), (c, e)}. Find \(R^{-1}\)?

If R is a relation from A to B where A and B are non-empty sets then inverse relation \(R^{-1}\) is given y: \(R^{-1}\) = {(b, a): (a, b) ∈ R}

⇒ \(R^{-1}\) = {(d, a), (e, c)}

9. Reflexive Relation: A binary relationship R specified on a set A is supposed to be reflexive if, for each element a ∈ A, we have aRa, that is, (a, a) ∈ R. This indicates that a relation specified on a set is a reflexive relation if and only if every component of the set is linked to itself. If there is a particular element of the set that is not related to itself, then the relation R is not a reflexive relation.

Example of reflexive relation: If the set B={3, 4, 5} then the relation {(3,3),(4,4),(5,5)} is a reflexive relation.

Here, each element of set B is mapped with itself, such that 3∈A, (3,3)∈R.

The number of reflexive relations on a set with n elements is presented by \(N=2^{n\left(n-1\right)}\), where capital N is the number of reflexive relations and small n is the number of elements in the set.

Representation of Types of Relations

Here is a brief summary of the types of relations in discrete mathematics along with their representation:

Some other types of relations depending on the mapping of two sets are as follows:

One-to-One Relations: A relation is said to be One to One relation if all the different elements of one set are outlined to different elements of another set.

One-to-Many Relations: A relation is supposed to be a One to Many relations if the identical element of one set is mapped to more than one element of a different set.

Many-to-One Relations: A relation is stated to be a Many-to-One relation if all the different elements of one set are mapped to the corresponding element of another set.

Many to Many Relations: A relation is supposed to be Many to Many relations if one or more than one component of one set is mapped with identical elements of another set.

Important Terms in Relations

We often address relations between any two or more objects. Below mentioned are some important terms regarding types of relations.

Domain of a Relation: Let R express a relation from set A to set B. Then, the set of all the first components of the ordered pair relating to relation R makes the domain of the relation R.

i.e. Domain (R) = {a: (a, b) ∈ R}

Example of relation: Let A = {2, 4} and B = {p, q} be two sets such that the relation R from A to B is given by:

R = {(2, p), (2, q), (4, p), (4, q)}.

Find the domain of given relation R.

The domain of a relation R is given by; Domain (R) = {a: (a, b) ∈ R}

⇒ Domain (R) = {2, 4}

Range of a Relation: Let R express a relation from set A to set B then the set of all second components of the ordered pair belonging to relation R forms the range of the relation R.

i.e. Range (R) = {b: (a, b) ∈ R}

Example of relation: If A = {a, c}, B = {d, e} and the relation R from A to B is given by: R = {(a, e), (c, e)}.

Find the range of given relation R.

The range of a relation R is given by Range (R) = {b: (a, b) ∈ R}

⇒ Range (R) = {e}

Codomain of a Relation: Let R signify a relation from set A to set B, then the set B is called the codomain of the relation R. i.e. Codomain (R) = B.

Representation of Relations

Relation defines the connection between two sets. There are 3 different methods of representing the relations. Each of them is discussed below with examples:

Representation of Relations in Set-Builder Form

A shorthand approach is applied to write sets and is often practiced for sets with an infinite number of components. It is practiced with various types of numbers, such as integers, natural numbers, whole numbers, real numbers and so on. The set-builder form is also applied to display sets with an interval or an equation.

Example: The relation of two sets P={3, 4, 5} and Q={9, 16, 25} in which elements of set P are the positive square root of components of set Q. This relation can be addressed in set-builder form as follows.

R={(a,b):a is the positive square root of b,a∈P,b∈Q,}

Representation of Relations in Roaster Form

Roster form is a description of a set that records down all of the elements existing in the set and is distributed by commas and confined within braces.

The relation can be expressed in roaster form by addressing all the possible ordered pairs of the two sets.

Example: The relation of two sets P={3, 4, 5} and Q={9, 16, 25} in which elements of set P are the square root of components of set Q. The relation can be formulated in roaster form as follows.

R = {(3,9),(4,16), (5,25)}

Representation of Relations in Arrow Diagram

In the arrow diagram method as the name suggests, the relation between sets is indicated by drawing arrows from the first elements to the second elements of all the combinations which relate to the relation.

That is the relationship between two sets is determined by using the arrow formed from one given set to another set.

Example: The relation of two sets P={3, 4, 5} and Q={9, 16, 25}, in which elements of set P are the square root of components of set the Q. The relation can be addressed by using an arrow diagram as shown below:

Important Points on Types of Relations

The important points on types of relations are given below:

• Relations and functions usually represent the various operations performed on sets.
• Relation in mathematics is defined as an association between the components of two or more sets, with a condition that the sets must be non-empty.
• A relation, say “R” is developed by a Cartesian product of subsets.
• Relations play a significant role in concepts like functional analysis and act as a base or foundation for many other areas in set theory.
• Relations, as well as functions, can be represented in various forms like set-builder form, arrow representation, algebraic form, graphic, roster form, and tabular form.
• All functions are relations but the converse that is all relations are functions is not true.
• Symmetric and anti-symmetric relations are not reverse of each other because a relation R can include both the properties or may not.
• A relation is said to be asymmetric if and only if it is both antisymmetric and irreflexive.
• The number of distinct relations from a set with “n” elements to a set with “m” elements is expressed by the formula \(2^{mn}\).
•  The number of reflexive relations on a set with n elements can be determined by the formula \(2^{n(n-1)}\).
• The number of symmetric relations on a set with n elements is determined by the formula \(2^{\frac{n\left(n+1\right)}{2}}\).
• The number of anti-symmetric relations on a set with n elements is determined by the formula \(2^n.3^{\frac{n\left(n-1\right)}{2}}\).

Applications of Different Types of Relations

Different types of relations like reflexive, symmetric, transitive, and equivalence are widely used in mathematics, computer science, and real-life modeling. They help in organizing data, building networks, defining hierarchies, and designing algorithms for sorting, scheduling, and classification.

 1. Reflexive Relations

• Application: In identity mapping, every element relates to itself.
• Example: In computer memory addressing, a location refers to itself.

2. Symmetric Relations

• Application: Used in undirected graphs and social networks where relationships are mutual.
• Example: If person A is a friend of B, then B is a friend of A.

3. Transitive Relations

• Application: Important in hierarchies, mathematical ordering, and permissions systems.
• Example: If A is older than B, and B is older than C, then A is older than C.

4. Equivalence Relations (Reflexive + Symmetric + Transitive)

• Application: Used to group elements into equivalence classes, common in modulo arithmetic, set partitioning, and geometry.
• Example: Two shapes are equivalent if they have the same area.

5. Partial Order Relation

• Application: Used in task scheduling, version control, and file organization.
• Example: In a company's employee hierarchy, if A supervises B, and B supervises C, then A supervises C indirectly.

6. Asymmetric Relations

• Application: Seen in tournaments and ranking systems, where the relationship goes in only one direction.
• Example: If team A defeats team B, then B cannot have defeated A in the same match.

7. Anti-symmetric Relations

• Application: Used in data structures and database design where unique hierarchy or ordering is required.
• Example: In ≤ (less than or equal to), if A ≤ B and B ≤ A, then A must be equal to B.

Solved Examples of Types of Relations

With the knowledge of composition of relations and their various types let us check out some of the solved examples for the same.

Example 1: Given a set X = {p, q, r}. Which of the following is an identity relation on X?

1) R = {(p, q), (p, r)}

2) R = {(p, p), (q, q), (r, r)}

3) R = {(p, p), (q, q), (p, r)}

4) none of these

Solution: Concept:

Identity Relation: Let A be a non-empty set then the relation \(I_{A}\) = {(a, a): ∀ a ∈ A} on A is called the identity relation on A.

Example:

If A = {1, 2, 3}, then relation given by the set {(1, 1), (2, 2), (3, 3)} is called the identity relation on the given set A denoted by \(I_{A}\).

Calculation: Given: X = {p, q, r}.

If A is a non-empty set then the relation \(I_{A}\) = {(a, a): ∀ a ∈ A} on A is called the identity relation on A.

So, the identity relation on a given set X is given by: \(I_{x}\)= {(p, p), (q, q), (r, r)}.

Hence option 2 is the correct one.

Example 2: Which of the following statements is/are TRUE for an empty relation R on a non-empty set S?

1) R is not reflexive

2) R is not symmetric

3) R is not transitive

4) All of these

Solution: Since R is empty and S is non-empty, R is not reflexive. Since R is empty, R is both symmetric and transitive. Hence option 1 is the correct one.

Example 3: How many reflexive relations are there on a set with 4 elements?

Solution: Data:

Number of elements in a set = n = 4

Formula: Total number of reflexive relations in a set = \(2^{n^{2}-n}\).

Calculation: Total number of reflexive relations in a set = \(2^{4^{2}-4}=2^{12}\).

Example 4: Consider the following statements:

(I) If the binary relation itself is transitive, then the transitive closure is that same binary relation.

(II) An asymmetric relation is always anti-symmetric.

(III) An empty relation on a nonempty set is always symmetric.

Which of the above statement/s is/are TRUE?

1) (II) and (III) only

2) (II) only

3) (I) and (III) only

4) (I), (II) and (III)

Solution: Answer: Option 4

Explanation:

(I) If the binary relation itself is transitive, then the transitive closure is that same binary relation, otherwise it will be different.

(II) An asymmetric relation is always anti-symmetric, as anti-symmetric relation allows self-loops in digraph but asymmetric does not. So the asymmetric condition is more strict.

(III) An empty relation on a nonempty set is always symmetric, but not reflexive.

Example 5: If the number of elements in a set A is 11, then how many anti-symmetric relations are possible on A?

1) \(2^{11}\)

2)\(2^{11}\times 3^{55}\)

3) \(11!\)

4) \(2^{11}\times 3^{11}\)

Solution: Number of anti-symmetric relations on a set with n elements =\(2^{n}\times3^{\frac{n^{2}−n}{2}}=2^{11}.3^{55}\).

Hence option 2 is correct.

Example 6: If A = {2, 4, 5}, B = {5, 9, 11} and R is the relation from A to B such that a R b ⇔ b = 2a + 1 then the inverse relation of R is?

Solution: Concept:

Inverse Relation: Let R be a relation from A to B. Then the inverse relation, denoted by \(R^{-1}\), from the set B to A, is defined as, \(R^{-1}\) = {(b, a) : (a, b) ∈ R}.

Calculation: Given: A = {2, 4, 5}, B = {5, 9, 11} and R is the relation from A to B such that a R b ⇔ b = 2a + 1.

So, when a = 2 ⇒ b = 2a + 1 = 5 ∈ B ⇒ (2, 5) ∈ R

When a = 4 then b = 2a + 1 = 9 ∈ B ⇒ (4, 9) ∈ R

When a = 5 then b = 2a + 1 = 11 ∈ B ⇒ (5, 11) ∈ R

⇒ R = {(2, 5), (4, 9), (5, 11)}

If R is a relation from A to B. Then the inverse relation, denoted by\(R^{-1}\), from the set B to A, is defined as, \(R^{-1}\) = {(b, a) : (a, b) ∈ R}.

⇒ \(R^{-1}\) = {(5, 2), (9, 4), (11, 5)}

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