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 cardinal number of power set of A is

**n[P(A)] = 2 ^{n}**

Note :

Cardinality of power set of A and the number of subsets of A are same.

**Example 1 :**

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)] = 2^{n}

Here "n" stands for the number of elements contained by the given set A.

The given set A contains five elements. So, n = 5.

Then,

n[P(A)] = 2^{5}

n[P(A)] = 32

Cardinality of the power set of A is 32.

**Example 2 :**

If the cardinal number of the power set of A is 16, then find the number of elements of A.

**Solution : **

The formula for cardinality of power set of A is given below.

n[P(A)] = 2^{n}

Here "n" stands for the number of elements contained by the given set A.

Then, we have

16 = 2^{n}

2^{4} = 2^{n}

4 = n

Number of elements of A is 4.

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"

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.

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).

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.

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**

Null set is a proper subset for any set which contains at least one element.

For example, let us consider the set A = {5}

It has two subsets. They are { } and {5}.

Here null set is proper subset of A. Because null set is not equal to A.

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.

**Example 1 :**

Let A = {1, 2, 3, 4, 5} and B = {5, 3, 4, 2, 1}. Determine whether B is a proper subset of A.

**Solution : **

If B is the proper subset of A, every element of B must also be an element of A and also B must not be equal to A.

In the given sets A and B, every element of B is also an element of A. But B is equal A.

So, B is the subset of A, but not a proper subset.

**Example 2 :**

Let A = {1, 2, 3, 4, 5} and B = {1, 2, 5}. Determine whether B is a proper subset of A.

**Solution : **

If B is the proper subset of A, every element of B must also be an element of A and also B must not be equal to A.

In the given sets A and B, every element of B is also an element of A and also But B is not equal to A.

So, B is a proper subset of A.

**Example 3 :**

Let A = {1, 2, 3, 4, 5} find the number of proper subsets of A.

**Solution : **

Let the given set contains 'n' number of elements.

Then, the formula to find number of proper subsets is

= 2^{n} - 1

The value of n for the given set A is 5.

Because the set A = {1, 2, 3, 4, 5} contains five elements.

Number of proper subsets = 2^{5} - 1

= 32 - 1

= 31

**Example 4 :**

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 = 2^{3} = 8.

Therefore,

P(A) = {{1}, {2}, {3}, {1, 2}, {2, 3}, {1, 3}, {1, 2, 3}, { }}

Apart from the stuff given above, if you need any other stuff in math, please use our google custom search here.

If you have any feedback about our math content, please mail us :

**v4formath@gmail.com**

We always appreciate your feedback.

You can also visit the following web pages on different stuff in math.

**WORD PROBLEMS**

**Word problems on simple equations **

**Word problems on linear equations **

**Word problems on quadratic equations**

**Area and perimeter word problems**

**Word problems on direct variation and inverse variation **

**Word problems on comparing rates**

**Converting customary units word problems **

**Converting metric units word problems**

**Word problems on simple interest**

**Word problems on compound interest**

**Word problems on types of angles **

**Complementary and supplementary angles word problems**

**Trigonometry word problems**

**Markup and markdown word problems **

**Word problems on mixed fractrions**

**One step equation word problems**

**Linear inequalities word problems**

**Ratio and proportion word problems**

**Word problems on sets and venn diagrams**

**Pythagorean theorem word problems**

**Percent of a number word problems**

**Word problems on constant speed**

**Word problems on average speed **

**Word problems on sum of the angles of a triangle is 180 degree**

**OTHER TOPICS **

**Time, speed and distance shortcuts**

**Ratio and proportion shortcuts**

**Domain and range of rational functions**

**Domain and range of rational functions with holes**

**Graphing rational functions with holes**

**Converting repeating decimals in to fractions**

**Decimal representation of rational numbers**

**Finding square root using long division**

**L.C.M method to solve time and work problems**

**Translating the word problems in to algebraic expressions**

**Remainder when 2 power 256 is divided by 17**

**Remainder when 17 power 23 is divided by 16**

**Sum of all three digit numbers divisible by 6**

**Sum of all three digit numbers divisible by 7**

**Sum of all three digit numbers divisible by 8**

**Sum of all three digit numbers formed using 1, 3, 4**

**Sum of all three four digit numbers formed with non zero digits**