Set Theory     

In our everyday life we often speak about collections of objects. 

A particular kind of sets such as, a football team, a group of students, bunch of flowers, etc.

In mathematics, we also come across the collections,

For example – Whole numbers, factors, natural numbers etc.

In simply we can say it’s a collection of things or numbers etc.

We examine the following collection,

1. The vowels in the English alphabet a, e, i ,o, u.

2. The natural numbers.

3. Even numbers 10 to 20.

4. States of India.

5. Solution  of a equation x + 3 = 5.

In above examples we noted that examples are well defined collection of objects, we can definitely decide whatever a given particular object belongs to a given collection or not.

For example we can say that the even number 24 does not belongs to the collection of even numbers 10 to 20, on the other hand, the number 18 belongs to the collection.

Again the collection of five greatest footballers of the world is not well defined, because the criterion of determining a footballer as must renowned may vary from person to person. or collection of beautiful flowers, are not well defined.

Thus, it is not a well defined collection of objects. so, we can say that a 

A set is a collection of different objects having some common properties. The collection could be anything like alphabets, numbers, pictures etc.

 “Set is a well defined collection of objects.”

Objects, Elements and Members are synonymous terms.

In a set we have list all objects of the collection, then enclose them in the “{ }” curly braces.

 Elements of a set

The different objects of a set are called the elements of a set.

Elements or members of a set are same thing. 

Generally, we denote a set with capital letters.

First we have to list all objects and separate them with comma and then enclose them in curly braces.

In the above figure a, b and c are the elements of a set so we have write them into brackets and the three dots represent that the number of elements are unlimited.

{a, b, c,…}

1. Sets are usually denoted by capital letters, A, B, C, D, E, F,…etc

2. The elements of a set are represented by small letters a, b, c, d, e, f,…etc.

If a is an element of a set A we say that 

“a belongs to set A” can be written as ‘a ∈ A’, and read as “a belongs to A”.  

If b is not an element of set A, we write b ∉ A and read as “b does not belong to A”.

Greek symbol ∈ (epsilon) is used to denote the “belongs to”.

In set theory we will learn about, 

Representation of sets by (statement form, roster or tabular form and Rule or Set builder form), 

Type of sets (finite and infinite sets, equal and equivalent sets, empty set, singleton set, cardinal number of a set universal set), subsets (Proper subset, super set, power set, proper subset), 

 Operation on sets(union, intersection, difference and complement sets)

Representation of a set

There are three methods of representing a set.

1. Statement form method

2. Roster or tabular form method

3. Rule or Set builder form method

  Statement form method

In this method the well defined description of the elements of the set is given, and the elements are enclosed in curly brackets { } .

For example –

1.  The set of whole numbers 29 to 35 is written as

             {Whole numbers 29 to 35} 

2. A set of singers of 21 years to 28 years.

Roster or tabular form method

1. In roster form we list each element or member, separated by a comma, and are enclosed some curly brackets { } around the whole thing.

For example – The set of natural numbers, form 1 to 5 is present in roster form as

                      {1, 2, 3, 4, 5} 

    set of 1 to 10 odd numbers form 1 to 10

                    {1, 3, 5, 7, 9} 

2.  In roster form we change the order of elements.

Thus the above set can also be written as

                    {3, 1, 7, 5, 9,}

The set of even natural numbers is written as

                   {2, 4, 6, 8, 10, 12…}

The dots represent that the list of even numbers continue indefinite.

3.When we write the set in roster form any element is not repeated.

    For example – The set of word “book’ is

                           {b, o, k} or {o, b, k,}

We know that, in roster form we change the order of elements.

  Rule or Set builder form

In this form all the elements of a set possess a single common property.

For example – In the set {a, e, i, o, u} all the elements possess a common property, each of them is a vowel in English alphabet, present the set by V we write
         V = {x: x is a vowel in English alphabet}

we describe the elements of the set by using a symbol ‘x’ or any other variable followed by colon “:”after the sign of colon we write the characteristic property possessed by the elements of the set and then enclosed the whole thing with brackets.

In this description the brackets stand for “the set of all” and the colon stands for “such that”. V = {x: x is a vowel in English alphabet}

The above example is read as the set of all x such that x is a vowel of English alphabet.

The symbol for special sets in mathematics are given below.

             Important sets in Mathematics 

1.  N = Set of all “Natural” numbers.

2.  W = Set of all “Whole” numbers.

3.  Q = Set of all “Rational” numbers.

4.   Z = Set of all “Integers”.

5.  R = Set of all “Real” numbers.

6.  Z ⁺  = Set of “Positive” Integers.

7.  Q ⁺ = Set of “Positive Rational” numbers.

8.  R ⁺  = Set of “Real” numbers.