Relation and Its Types

Introduction

In mathematics, the concept of relations and its types is a fundamental part of set theory. A relation in math describes how the elements of the sets are related to each other, i.e., it states the connection between two sets. There are different types of relations with specific properties and characteristics. Learning more about relations and their types – such as reflexive, symmetric, transitive, and equivalence relations – is crucial for solving problems in mathematics, computer science, and logic. Let’s start by understanding the fundamentals of relations by exploring more about what is a set, types of sets, types of relations, and how these concepts are interconnected.

Table of Content

• Definition
• Introduction to Sets
• Types of Sets
• Relations & its Types
• Real-life Applications of Sets, Relations, and its Types
• Representation of Types of Relations
• Conclusion
• Frequently Asked Questions on Sets
  
  

Definition

Introduction to Sets

A set is a collection of elements or members, where members can represent things like numbers, animals, days, months, etc. For example, the set of the first five even numbers is represented as A = {2, 4, 6, 8, 10}, where 2, 4, 6, 8, and 10 are called the elements of the set ‘A’. We use the symbol '∈' to show that an element belongs to a set, i.e., 4 ∈ A

There are three different forms in which a set can be represented:

1. Semantic form
2. Roster form
3. Set builder form

 

The connection between the members of two sets is called a relation. For example, the students standing in ascending order in morning assembly is an order relation between students and their height.

 

Types of Sets

Learning about types of sets will help us understand how to define and analyze sets as well as relations. By practicing sets questions, you can easily grasp the concept clarity required for this topic.

•  Empty Set: A set with zero or no elements is called an empty set. For example, A={}
•  Singleton Set: A singleton set is a set with a single or only one element. For example, A={1}
•  Finite Sets: A finite set is a set with a countable number of elements. 
•  Infinite Sets: An infinite set is a set of an uncountable number of elements.
•  Subset and Superset: When one set is a subset of the other. Relationships between two sets.
•  Universal Set: A set containing all possible elements under consideration.

 

Relations and its Types

There are 8 types of relations listed as under:

1. 1. Empty Relation

   2. Universal Relation

   3. Identity Relation

   4. Inverse Relation

   5. Reflexive Relation

   6. Symmetric Relation

   7. Transitive Relation

   8. Equivalence Relation

 

Empty Relation:

When no element of the set is mapped to the elements of another set or itself, it is known as a void or empty set. An empty set is non-reflexive and is represented as R = ∅

Universal Relation:

When all elements of a set are related to the elements of the same set or another set, it is a universal relation. It is the opposite of an empty relation. For example, if A={a, b}, then R= {(a,a), (a,b), (b,a), (b,b)}

Identity Relation:

An identity relation is a relation in which each element is related to itself only. For example, if A={1,2}, then R={(1,1), (2,2), (3,3)}

Inverse Relation:

An inverse relation is when a set has elements that are inverse elements of the other. For example, if A = {(a, b), (c, d)}, then the inverse relation will be R-1 = {(b, a), (d, c)}. So, for an inverse relation, R⁻¹ = {(b, a): (a, b) ∈ R}

Reflexive Relation:

A set is reflexive when every element of the set maps to itself. For every a belonging to R, there is a pair (a,a) that belongs to R.

Symmetric Relation:

A relation is said to be symmetric when (b, a) ∈ R is true when (a, b) ∈ R. For example, a symmetric relation will be R = {(1, 2), (2, 1)} for a set A = {1, 2}.

Transitive Relation:

For a transitive relation, if (x, y) ∈ R and (y, z) ∈ R, then (x, z) ∈ R. For a transitive relation, aRb and bRc ⇒ aRc ∀ a, b, c ∈ A

Equivalence Relation:

When a relation is reflexive, symmetric, and transitive at the same time, it is known as an equivalence relation. Some of the example of equivalence relation in math is equal to ‘=’

Representation of Types of Relations

 

 

Real-life Applications of Sets, Relations, and its Types:

The concept of sets and relations holds significance in our day-to-day life, from organizing kitchen cabinets to listening to our favorite playlist. The grocery list that we usually create is a universal set of kitchen items, which further contains the subsets of similar items like fruits, vegetables, dairy, etc. Restaurant menus, jewelry boxes, makeup kits, and supermarkets are some of the more common examples of sets in real life. Understanding the fundamentals of sets is important to organize and understand the world around us.

 

Frequently Asked Questions on Sets

1. What are sets?

Answer: A set is a collection of elements or members, where members can represent things like numbers, animals, days, months, etc. 

2. What is a relation in math?

Answer: A relation in math describes how the elements of the sets are related to each other, i.e., it states the connection between two sets.

3. Name types of relations.

Answer: (1) Empty Relation (2) Universal Relation (3) Identity Relation (4) Inverse Relation (5) Reflexive Relation (6) Symmetric Relation (7) Transitive Relation (9) Equivalence Relation

4. What is a Venn diagram?

Answer: A Venn diagram is a diagram that enables us to visualise the logical relationship between sets & their elements.
 

5. What are the types of relations based on the mapping of two sets?

Answer: The types of relations based on the mapping of two sets are one-to-one, one-to-many, many-to-one and many-to-many.