Sets-1st-class.10-mathematics-introduction-definition of set-roster form-set builder form-belongs to-cardinal number of set

 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 }

Example.2:

Prime numbers less than 12.

we can answer easily for this also

Because it is well defined

we write the set

 { 2, 3, 5, 7, 11 }

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 is the Set of days in a week

    

    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"

-----------------------------------

Example.9: 

B = { 1, 2, 3, 6 }

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"

-----------------------------------

Example.10:

 K = { -1, 0, 1, 2}

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"