Real life Analogy:

Each book is an element in the "book set." 

Defination of set.

Set is a well-defined collection of objects or people.

Ex: 1. Number of colours in a rainbow.

      2. The collection of vowels of the English Alphabet.

      3. The collection of natural numbers between 30 and 35.

Representation of sets.

Sets can be represented in two ways:

Roster form

Defination: In roster form, all the elements of the set are listed, separated by commas and enclosed between curly braces { }. 

Example: If set represents all the leap years between the year 1995 and 2015, 

then it would be described using Roster form as:

A ={1996,2000,2004,2008,2012}

Problems

Que: Write the following sets in roster form:

(i) A = {x: x is an integer and –3 < x < 7}.

    Solution:

    A = {–2, –1, 0, 1, 2, 3, 4, 5, 6}

(ii) B = {x: x is a natural number less than 6}

     Solution:

     B = {1, 2, 3, 4, 5}

Set builder form

Defination: In set builder form, all the elements have a common property. This property is not applicable to the objects that do not belong to the set.

Example: If set S has all the elements which are even prime numbers, it is represented as:

S={ x: x is an even prime number}

where ‘x’ is a symbolic representation that is used to describe the element.

‘:’ means ‘such that’

‘{}’ means ‘the set of all’

Cardinality of set

Defination: The cardinality of a set is the number of elements in the set.

It's also known as the size of the set. 

Example: If A = {a, b, c, d, e}, 

                   then n(A) (or) |A| = 5

Problems

Que: Find cardinality of following sets.

(i)  Let P = {k, l, m, n}. 

    Solution:

     n(P) = 4.

(ii) B = {2, 4, 6, 8}. 

     Solution:

     n(B) = 4.

Types of sets

1.Finite set:

A finite set is a set that has a finite number of elements.

Example: A={1,3,5,7} is a finite set with four elements. 

2. Infinite set:

An infinite set is a set whose elements can not be counted or an infinite set is one that has no last element.

Example: W = {0, 1, 2, 3, ..} i.e. set of all whole numbers is an infinite set.

Venn diagrams

Venn diagrams are the diagrams that are used to represent the sets, relation between the sets and operation performed on them, in a pictorial way. 

Defination:

Union of sets

Defination: 

Union of two or more sets is the set containing all the elements of the given sets.

Union of sets can be written using the symbol “⋃”

X ∪ Y = {a: a ∈ X or a ∈ Y}

Problem

Que: Let U be a universal set consisting of all the natural numbers until 20 and set A and B be a subset of U defined as A = {2, 5, 9, 15, 19} and B = {8, 9, 10, 13, 15, 17}. Find A ∪ B.