POWER SET OF A SET
About "Power set of a set"
Power set of a set :
The set of all subsets of A is said to be the power-set of the set A.
Reading notation :
The power-set of A is denoted by P(A).
Power-set of a set - Examples
Example 1 :
Let A = {1, 2, 3 } find the power-set of A.
Solution :
We know that the power-set is the set of all subsets.
Here, the given set A contains 3 elements.
Then, the number of subsets = 2³ = 8
Therefore,
P(A) = { {1}, {2}, {3}, {1, 2}, {2, 3}, {1, 3}, {1, 2, 3}, { } }
Example 2 :
Let A = {a, b, c, d, e} find the cardinality of power-set of A
Solution :
The formula for cardinality of power-set of A is given below.
n[P(A)] = 2ⁿ
Here "n" stands for the number of elements contained by the given set A.
The given set A contains "5" elements. So n = 5.
Then, we have
n[P(A)] = 2⁵
n[P(A)] = 32
Hence, the cardinality of the power-set of A is 32.
Apart from the stuff "Power-set of a set", let us come to know some other important stuff about subsets of a set.
Subset of a given set
A set X is a subset of set Y if every element of X is also an element of Y.
In symbol we write
x ⊆ y
Reading Notation :
Read ⊆ as "X is a subset of Y" or "X is contained in Y"
Read ⊈ as "X is a not subset of Y" or "X is not contained in Y"
Proper subset
A set X is said to be a proper subset of set Y if X ⊆ Y and X ≠ Y.
In symbol, we write X ⊂ Y
Reading notation :
Read X ⊂ Y as "X is proper subset of Y"
The figure given below illustrates this.
Super set
A set X is said to be a proper subset of set Y if X ⊆ Y and X ≠ Y.
In symbol, we write X ⊂ Y
Here,
Y is called super set of X
Formula to find number of subsets
If A is the given set and it contains "n" number of elements, we can use the following formula to find the number of subsets.
Number of subsets = 2ⁿ
Formula to find the number of proper subsets :
Number of proper subsets = 2ⁿ⁻¹
Cardinality of power set
We already know that the set of all subsets of A is said to be the power set of the set A and it is denoted by P(A).
If A contains "n" number of elements, then the formula for cardinality of power set of A is
n[P(A)] = 2ⁿ
Note :
Cardinality of power set of A and the number of subsets of A are same.
Null set is a subset or proper subset
Null set is a proper subset for any set which contains at least one element.
For example, let us consider the set A = { 1 }
It has two subsets. They are { } and { 1 }.
Here null set is proper subset of A. Because null set is not equal to A.
If null set is a super set
If null set is a super set, then it has only one subset. That is { }.
More clearly, null set is the only subset to itself. But it is not a proper subset.
Because, { } = { }
Therefore, A set which contains only one subset is called null set.
After having gone through the stuff given above, we hope that the students would have understood "Power-set of a set".
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