Sets in Mathematics, Example of Sets, Subset, Basic Set Theory

Sets in Mathematics

In This topic you will learn about Sets in Mathematics, Example of Sets, Subset, Basic Set Theory, Definition of Set, Element, Notation of Set, Examples of notation, Representation of Sets, Types of Set

Definition of Set:-

Set is a well defined collection of object.

Meaning of well Define:- Well defined means object will surely belong to that collection…

  1. Set may consist same type of objects like a collection of natural numbers is a set.
  2. Set may consist different type of objects like a pen, a book, a box lying on table.

Element:- The object which belongs to a set  is called element. Element is also called member.

Notation of Set:-

  1. Sets are denoted by Capital letters A,B,C etc.
  2. Elements of set or denoted by small letters a,b,c etc.

Let ‘A’ be any set and ‘a’ is member of set ‘A’ then it is written as –  a£A

It means ‘a belongs to A’    Or  ‘a is an element of A’  Or ‘a is a member of A’

If ‘a’ is not a member of ‘A’ then it is written as –  a€A

Examples of notation:-

  • N : the  set of all natural numbers
  • Z : the set of all integers
  • Q :  the set of all rational numbers
  • R : the set of all real numbers

Representation of Sets:-

A set is expressed by two ways.

  1. Tabular or Roster Form
  2. Set-builder Form or Rule Method

1.Tabular Form:-

In this form all the elements of set are separated by commas and enclosed in curly brackets { }

Example:- A set consists numbers 1,2,5,7,9 then it is written in Tabular or Roster Form as:-

{1,2,5,7,9}

Elements can be written in any order like

{2,5,1,7,9,}  or {1,5,2,9,7}

 

2. Set Builder Form:-

In this form Set is written by some special property which is satisfied by all the elements of the set.

Example:- A set has elements a,e,i,o,u

*To write this set in set builder form,we have to see some property which will satisfy by these all elements…

Here we see Set has elements a,e,i,o,u… These are vowels…Hence it is represented as-

A={x:x is vowel in English alphabets}

Types of Set:-

  1. Finite Set
  2. Infinite Set
  3. Singleton Set
  4. Empty Set
  5. Equal Sets
  6. Equivalent Sets

1. Finite Set:-

If the elements of a set are finite in number, then the set is called finite set.

Example:-

  • The set of days in a week.
  • The set of boys in all Delhi schools.

2. Infinite Set:-

If the elements of a set are infinite in number,then the set is called infinite set.

Example:-

  • The set of natural numbers like {1,2,3,4,5,….}
  • The set of whole numbers like {0,1,2,3,4,5,……}

3. Singleton Set:-

A set which consists only one element is called Singleton set.

Like {2} is a singleton set.

4. Empty Set:-

A set which consists no element is called Empty set.

Empty set is also called “Null Set” or “Void Set”.

It is denoted by the symbol{ }.

Examples:-

The set of all even integers greater than 8 and less than 10.

It will be Empty Set because between 8 and 10 there us only odd integer.

5. Equal Sets:-

Two sets A and B are said to be equal if every element of A is an element of B and every element of B is an element of A.

Like if a£A then a£B

And

If b£B then b£A

 

Example-

{1,3,7,9}={3,1,9,7}

6. Equivalent sets:-

Two sets A and B are said to be equivalent sets if the number of elements in set A is equal to the number of elements in set B.

n(A)=n(B)

It is denoted by the symbol ‘~’.

A={1,2,5} and B={a,b,c} A and B are equivalent sets because the number of elements in both sets are same.

we write A~B . It is read as ‘A is equivalent to B’.

 

Cardinal Number of a Finite Set:-

The number of elements in a finite set is called “Cardinal Number” Or ” Order of a Finite Set”

A={1,2,4,5} then n(A)=4

Here ‘4’ is cardinal number of Set ‘A’.

 

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