What is Set , Definition of Set , Elements of Set , Notation, Representation , Properties of set and Examples on Sets
The set in mathematics is the group of objects and the objects should be distinct. Repetition of objects is not allowed in sets.
The sets are used to define the concepts of relations and functions. the theory of sets is developed by German mathematician Georg Cantor (1845-1918).
Definition of Set
A set is any collection of distinct in simple way set is a collection of objects specified in such a way that we can determine whether a given object is present in the set or not.
for example :
1. A set of vegetables
2. A set of numbers
3. Set of flowers
4. A set of keys
Elements of a set
The different objects or members present in the set are called as elements of set. The elements of a set are written in any order . the elements present in the set should not be repeated. the elements are denoted by small letters.
A set is usually denoted by capital letters and the elements of the set are denoted by small letters.The elements of the set are enclosed with curly brackets.({})
For example :
- A ={1,2,3,4,5}
- B ={a,s,d,f,g,j}
there are some standard sets :
|
Description |
Set |
|
The set
of natural numbers |
N = {1,2,3,4,5,……..} |
|
The set
of whole numbers |
W= {0,1,2,3,4,5,….} |
|
The set
of integers |
Z or I
={….-3,-2,-1,0,1,2,3,…..} |
|
The set
of even numbers |
E = {2,4,6,8,10,……} |
|
The set
of odd numbers |
O =
{1,3,5,7,9,……} |
methods to denote a set
there are three methods to represent a set they are as follows:
- description
- Roster form or Listing method
- Set builder form or Rule method
3.Set builder form : Set builder form is also known as Rule method .
Sr.NO. | Description | Roster form | Set builder form |
1 | A set of
first ten natural numbers | A={1,2,3,4,5,6,7,8,9,10} | A={x|x is a
natural number less than 11} |
2 | A set of vowels
| J={a,e,i,o,u} | J={x|x is a
vowel } |
3 | A set even
numbers less than 10 | P={2,4,6,8} | P={x|x is an
even natural number less than 10} |
4 | A set of
letters in the word ‘set’ | W={s,e,t} | W={x|x is a
letter in the word set} |
Properties of sets:
1. The change in the order of set elements does not make any change in
the set .
If we have two sets like:
A = {1,3,5,4,2}
B = {1,2,3,4,5}
The above two sets are equal
means they are the same. Because, the elements present
In both the sets is same. i.e., the elements
which are present in set A are present in set B.
Similarly,
{a,e,i} = {i,e,a} =
{e,i,a}
2. If one or
more elements in the set are repeated, the set remains same.
In the other words the elements of the set
should be distinct.
A =
{1,1,2,2,3,4,5,5,5,4,4,4} and B = {1,2,3,4,5}
Both are same.
Some more symbols which are used in sets :
‘∈’ this symbol
is used to denote elements and their related set
'∉' this symbol is used to denote elements and their
related set
For example:
S={ a,b,c,d,e }
a,b,c,d,e are the elements which are present in the set and it is
represented as
a ∈ A
b ∈ A
c ∈ A
d ∈ A
e ∈ A
and the
elements which doesnot belongs to the set is represented as
Cardinality:
The cardinality of a set is the total number of elements present in the set .
For example:
A={1,2,3,4,5,6}
The cardinality of the set A is 6 and it is represented as:
| A | = 6 or n(A) = 6
The cardinality of set A is equal to five . Symbolically,
|A| = 5
here in the picture ,
the set of four bit-root is shown , here the cardinality of the set of bit-root is four
B={a,e,i,o,u,z,s,t}
The cardinality of set B is equal to 8 . Symbolically,
|B| = 8
Examples:
1. write the set of odd numbers less than 7 in roaster form and in set builder form .
in roaster form :
B={1,3,5}
in set builder from :
B={x|x is a odd number less than 7}
2. Write the cardinality of the following sets:
i) A={1,2,3,4,5,6,7,8}
cardinality of A is 8 symbolically,
|A| = 8
ii) G={a,s,e,d,f,g}
cardinality G of is 6 symbolically,
|G| = 6
Operations-on-sets Union-Intersection-Difference-Complement and The Laws Of The Operations
Venn Diagram, Examples on venn diagram
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