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 its 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 sets, relations & its types by exploring more about the types of sets, types of relations, and how these concepts are interconnected.
Table of Content
Set: A set is simply a well-organized collection of similar objects or elements. It is often represented using curly brackets. For example, A = {2, 4, 5, 6….}
Relations: A relation is a rule that defines the connection between the 2 sets. We will learn more about relations and functions along with their types later.
Function: A function is defined as a relation in which each element of a set is associated with an element of the other.
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:
Semantic Form |
Roster Form |
Set builder form |
Set of first five even natural numbers |
{2, 4, 6, 8, 10} |
{x ∈ A | x ≤ 10 and x is even number} |
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.
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.
There are 8 types of relations listed as under:
Empty Relation
Universal Relation
Identity Relation
Inverse Relation
Reflexive Relation
Symmetric Relation
Transitive Relation
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 ‘=’
Relation Type |
Condition |
Empty Relation |
R = ∅ |
Universal Relation |
R= {(a,a), (a,b), (b,a), (b,b)} |
Identity Relation |
I = {(a, a), a ∈ A} |
Inverse Relation |
R-1 = {(b, a): (a, b) ∈ R} |
Reflexive Relation |
(a, a) ∈ R |
Symmetric Relation |
aRb ⇒ bRa, ∀ a, b ∈ A |
Transitive Relation |
aRb and bRc ⇒ aRc ∀ a, b, c ∈ A |
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.
Answer: A set is a collection of elements or members, where members can represent things like numbers, animals, days, months, etc.
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.
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
Answer: A Venn diagram is a diagram that enables us to visualize the logical relationship between sets & their elements.
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.
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