10th class-mathematics-Sets
Types of Sets
1. Null Set or Empty Set
A Set Without elements is called Null Set
Null Set is denoted by { }
The symbol "Ø" is used to represent a null set
This is read as "pi"
Cardinal number of Null Set is "ZERO"
i.e. n(Ø) = 0
Example.1:
A is the set of months of a year having 35 days
No months in a year with 35 days
So A is a Null set
∴ A = { }
and cardinal number of A is "zero"
∴ n(A) = 0
Example.2:
C is the set of composite numbers less than 4
Composite numbers are start from 4
That means 4 is the smallest composite number
So there is no composite numbers less than 4
Therefore C is a null set
∴ C = { }
and
n(C) = 0
Note:
Ø, { 0 } , { Ø } are three different sets
Ø is a null set
{ 0 } is a set with element "0"
{ Ø } is set with element "Ø"
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2. Singleton Set
If A Set contained only One element, It is called Singleton Set
Example.3:
P is the set of even prime numbers
Here even prime number is only 2
So, P = { 2 }
And n(P) = 1
Here P is called "Singleton Set"
Example.4:
X is the set of Integers neither negative nor positive
In integers only "0" is not negative and not positive
So, X = { 0 }
And n(X) = 1
Here X is called "Singleton Set"
Note:
Cardinal
number of a Singleton Set is ONE.
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3. Finite Set
If a Set contained finite number of elements, it is called Finite Set.
In a Finite set elements are countable(limited).
That means we can count the elements in the set
Note:
Cardinal number of a
Finite Set is Whole number.
Example.5:
D is the set of days in a week
∴ D = { Sun, Mon, Tue, Wed, Thu, Fri, Sat }
Elements in the set D are Countable
And n(D) = 7
So, D is a finite set
Example.6:
F is the set of factors of 24
∴ F = { 1, 2, 3, 4, 6, 8, 12, 24 }
Elements in the set F are Countable
And n(F) = 8
So, F is a finite set
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If a Set contained infinite number of elements, It is called Infinite Set.
Infinite set has unlimited elements.
That means we can not count the elements in the set
A set which is not finite set is called an Infinite set
Note:
Cardinal number of an Infinite Set is not defined.
Examples.7:
S is the set of stars in
the Sky
we can't count the stars in the sky
they are unlimited
So, set "S" is an Infinite set
Examples.8:
P is the set of points on a line
we can't count the points on the line
those are unlimited
So, set "P" is an Infinite set
Examples.9:
K is the set of integers less than 5
∴ K = { 4, 3, 2, 1, 0, -1, -2, -3, -4, -5, -6,............... }
Elements in the set are unlimited
So, set "K" is an Infinite set
M is
the set of multiples of 3
∴ M = { 3, 6, 9, 12, 15, 18,................. }
Elements in the set are unlimited
So, set "M" is an Infinite set
Two sets are equivalent, if their number of elements(cardinal number) are same
Elements in equivalent sets could be same or different,
But cardinal number of sets should be same
Example.12: A={ 2, 3, 6, 8 } and B={ 1,
2, 3, 4 }
Here n(A)=
4 and
n(B)=4
Elements are different in both sets
But cardinal numbers are equal
∴ A , B are Equivalent
sets
Elements are different in both sets
But cardinal numbers are equal
Example.13:
M={ a, b, c, d, e, f } and N={ 2, 3, 6, 9, 10, 15 }
Here n(M)= 6 and n(N)=6
But cardinal numbers are equal
So, M and N are Equivalent sets
Example.14:
X={ 0, 3, 6, 9 } and Y={ 3, 6, 9 }
Here n(X)= 4 and n(Y)=3
And cardinal numbers are unequal
So, X and Y are not Equivalent sets
Two sets are said to be equal sets, if their elements exactly equal
Here elements in both the sets are equal
And cardinal numbers are also equal
If sets A and B are equal sets,
we write them as
A=B
Example.15:
Elements are equal in above two sets
And n(A)=4 and n(B)=4
So, A and B are equal sets
Example.16:
C={ a, b, c, d, e } and D ={ c, b, e, d, a }
Elements are equal in above two sets
And n(C)=5 and n(D)=5
∴ C and D are equal sets
∴ C=D
Example.17:
n(E)=4 and n(F)=4
But Elements are not same in above sets
So,
E and F are not equal sets
∴ E≠F