Mathematics for class 10- Sets-Introduction of Sets
Examine the following statements
1. All Students in your class
2. All States in India
3. Natural numbers greater than three and less than twelve
4. Five good books in your school library
5. Three best singers in India
First three statements are easy to answer
But last two statements are difficulty
And any two persons may not give same answers for last two questions
Because first three statements are defined correctly to answer everyone "same"
But not last two statements.
Definition of Set
A Set is a well defined collection of distinct Objects or Numbers or Things or Ideas
Objects in the Set are called Elements
A Set is said to be well defined,
If we can decide Any object from the universe is an element of the set or not.
Elements of a Set are written inside a pair of curly braces separated by commas.
Elements can’t be repeat in a set
That means a set contains distinct(different) elements
Example.1:
Vowels in English alphabet.
we can answer easily for this(Why)
Because it is well defined
we write the set
{ a, e, i, o, u }
we can answer easily for this also
Because it is well defined
we write the set
Note:
A set is named by Capital letters of English alphabet
And
An element is denoted by small letters of English alphabet
Example.3:
A is the set of days which are starts with “T”.
A = { Tuesday, Thursday }
Example.4:
M is the set of letters in the word “MATHEMATICS”.
M = { m, a, t, h, e, i, c, s }
Roster form of a set
If a set is written by its
“All elements inside of Flower brackets with using commas”,
we say that the Set is in “Roster form”
Note:
No preference is given to Order of elements in a set
Example.5:
A= { Sun, Mon, Tue, Wed, Thu, Fri, Sat}
Example.6:
R is the set of Alphabet in the word INDIA
R= {i, n, d, a }
or
R= {a, d, i, n }
Here both sets are equal
not given any priority to the order of elements in the set
Example.7:
P is the set of prime numbers less than 10
P= { 2, 3, 5, 7 }
Set builder form of a set
If A set is written by “common property” of its elements, we say that the set is in “set builder form”
Note:
we can write set builder form of a set in "different ways"(Based common property of its elements)
Example.8:
A = { 2, 4, 6, 8 }
Elements in the set A are "Even numbers"
Even numbers is common property of elements
and Even numbers are less than 10
So we write set builder form as
A= {x/x is even number and x<10 }
we read this "A is the set of x such that x is even number and x less than 10"
OR
Elements in the set A are "multiples of 2"
"Multiples of 2" is common property of elements
So we write set builder form as
A= { 2x/x is natural number and x<5 }
we read this "A is the set of 2x such that x is natural number and x less than 5"
-----------------------------------
Do you know the relation among the elements?
Think about the relation
All the elements in the set B are " factors of 6"
"Factors of 6"is the common property of elements
So we write set builder form as
B= {x/x is a factor of 6 }
we read this "B is the set of x such that x is a factor of 6"
-----------------------------------
Do you know the common property of elements?
Think about common property
The elements are integers and lie between -2 and 3
"Integers" is the common property
So we write set builder form as
K= {x/x is integer and -1 ≤ x ≤ 2 }
we read this "K is the set of x such that x is an integer and -1 is less than or equal to x and x less than or equal to 2"
Belongs to(∈)
The symbol “∈” is used for the words “ belongs to ” or “is an element of”
Does not Belongs to(∉)
The symbol “∉” is used for the words “Does not belongs to”or “ is not an element of”
Example.11:
A = { 1, 3, 7, 8 }
1 is an element in set A
So we say "1 is belongs to A"
We write as
1 ∈ A
Read as "1 belongs to A"
-----------------------------------
2 is not an element in set A
So we say "2 is not belongs to set A"
We write as
2 ∉ A
Read as "2 does not belongs to A"
-----------------------------------
Cardinal number of a Set
The number of elements in a set is called the Cardinal number of the Set.
Cardinal number of Set A is denoted as n(A).
Read as "number of A"
Example.12:
A = { a, e, i, o, u }
set A has only 5 elements
So
Cardinal number of Set A is 5
∴ n(A) = 5
-----------------------------------
Example.13:
B = { 4, 8, 12 }
Set B has only 3 elements
So
Cardinal number of Set B is 3
∴n(B) = 3
Click for "worksheet on above concepts"
Click the link for "Types of sets, Basic set operations and Venn diagrams"
Click the link for "Euclid division algorithm&HCF , Proof of Irrational number"