Chapter Checklist »
✓ Sets and Their Types
✓ Subsets of Set
✓ Venn Diagrams and Operations on Sets
✓ Applications of Set Theory

The set theory was developed by German Mathematician Georg Cantor (1845-1918). He first encountered sets while working on ‘problems on trigonometric series’. This concept is used in every branch of Mathematics i.e., Geometry, Algebra, etc.

» Topic: Sets and Their Types

In our daily life, while performing our regular Work, we often come across a variety of things that occur in groups.

e.g., Team of cricket players, group of tall boys, group of teachers, etc.
The words used above like team, group, etc., convey the idea of certain collections.

What is a set?

A set is a collection of objects. The objects in the set are called elements of the set. A set is well-defined if there is a way to determine if an object belongs to the set or not. To indicate that We are considering a set, the objects (or the description) are put inside a pair of set braces, {}.

📌 Example 1. Are the following sets well—defined?
(i) The set of all groups of size three that can be selected from the members of this class.
(ii) The set of all books Written by John Grisham.
(iii) The set of great rap artists.
(iv) The best fruits.
(v) The 10 top—selling recording artists of 2018.
✍ Solution:
(i) You can determine if a group has three people and whether or not those people are members of this class so this is well—defined.
(ii) You can determine whether a book was written by John Grisham or not so this is also a Well—defined set.
(iii) A rap artist being great is a matter of opinion so there is no way to tell if a particular rap artist is in this collection, this is not well—defined.
(iv) Similar to the previous set, best is an opinion, so this set is not well—defined.
(v) This is well—defined, the top selling recording artists of any particular year are a matter of record.

Well-defined Collection of Objects

If any given collection of objects is in such a way that it is possible to tell without any doubt whether a given object belongs to this collection or not, then such a collection of objects is called a well-defined collection of objects. e.g., ‘The rivers of lndia’ is a Well—defined collection. Since, We can say that the river Nile does not belong to this collection. On the other hand, the river Ganga does belong to this collection.

Here, a group of intelligent students, a group of most talented writers of Canada, group of pretty girls are not Well-defined collections, because we can not decide Whether a given particular object belongs to the given collection or not.

📌 Example 2. Which of the following are sets? Justify your answer.
(i) The collection ofall the months ofa year beginning with the letter J.
(ii) The collection often most talented writers of Canada.
(iii) A collection of novels written by the writer Munshi Prem Chand.
(iv) A collection of most dangerous animals of the world.
✍ Solution:
(i) We are sure that members of this collection are January, June and July.
So, this collection is well—defined. Hence, it is a set.
(ii) A writer may be most talented for one person and may not be for other. Therefore, we cannot definitely decide which writer will be there in the collection.
So, this collection is not well—detined. Hence, it is not a set.
(iii) Here, we can definitely decide whether a given novel belongs to this collection or not.
So, this collection is well—defined. Hence, it is a set.
(iv) The term most dangerous is not a well—defined term. An animal may be most dangerous for one person and may not be for the other.
So, it is not well—defined. Hence, it is not a set.

• Sets and Their Types — Types of Sets