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Representation of Sets

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Representation of Sets - Lesson Summary

A set is a well-defined collection of objects.

By well-defined collection, we mean that we should be able to decide whether a particular object belongs to a definite collection or not.
Example of a well-defined collection is natural numbers less than five. This collection of natural numbers is well defined, as we can definitely decide whether a given particular number belongs to this collection or not.

An example of a set that is not well defined is a set that consists of clever students. This is a not a well-defined set because the choice of clever students is not clear, since there is no accurate measure for cleverness.

A set is denoted with a capital letter.
E.g.: A, Q, S

The elements of a set are denoted with small letters.
e.g.: x, p, s, etc.

The word element is used synonymously with the words object and member.
A= {p, q, r}

The element p belongs to set A.

p ∈ A
z ∉ A

N : The set of natural numbers

Z : The set of integers

Q : The set of integers

R : The set of real numbers

Z+ : The set of positive integers

Q+ : The set of positive rational numbers

R+ : The set of positive real Numbers

There are two ways of representing sets.
1. Roster form
2. Set-builder form

Roster form
This method of representing sets is also called the tabular form.
Ex: Prime numbers less than 10.

P = {2, 3, 5, 7}
Similarly, a set of natural numbers between 20 and 30 is represented as:

A = {21, 22, 23, 24, 25, 26, 27, 28, 29}
The order of the elements in a set is unimportant.

While representing a set in roster form, the elements are generally not repeated. Only distinct elements are written while representing a set in roster form.

Set-builder form
The set-builder form is a more concise form of a given set.
To write the set in the set-builder form, start with a bracket and write the variable.
Then a colon is placed and a property is assigned to this number.
Ex 1: {2, 3, 5 and 7} is written in set-builder form as P = {x:x is a prime number, x < 10}.
The set is named as P and read as P is the set of all x such that x is a prime number less than 10.
Ex 2: Represent the set of natural numbers between 20 and 30 using the set-builder form.
Set of natural numbers between 20 and 30 = {21, 22, 23, 24, 25, 26, 27, 28, 29}

We name this set A.
A = {n:n is a natural number, 20 < n < 30}
This is read as: A is the set of all n such that n is a natural number and n is lies between 20 and 30.


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