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).

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 five elements. So n = 5. 

Then, we have 

n[P(A)] = 2⁵

n[P(A)] = 32

The cardinality of the power set of A is 32. 

Subset of a 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.

List the elements of the set A, which is the set of all:

Problem 1 :

positive whole numbers between 5 and 10

Solution :

A = {6, 7, 8, 9}

Problem 2 :

odd numbers between 10 and 20

Solution :

A = {11, 13, 15, 17, 19}

Problem 3 :

months of the year

Solution :

A = {January, February, March, April, May, June, July, August, September, October, November, December}

Problem 4 :

factors of 30

Solution :

A = {1, 2, 3, 5, 6, 10, 15, 30}

Problem 5 :

planets of the solar system

Solution :

A = {Mercury, Venus, Earth, Mars, Jupiter, Saturn, Uranus, and Neptune}

Problem 6 :

letters which make up the word BASEBALL

Solution :

A = {B, A, S, E, L}

Problem 7 :

colours of keys on a piano

Solution :

A = {C, D, E, F, G, A, B}

Problem 8 :

square numbers between 50 and 60.

Solution :

There is no square number in between 50 and 60. Then there must be a null set.

A = { }

Problem 9 :

Find n(A) for each of the sets

Let P = {1, 5, 7, 8, 10} and Q = {1, 4, 5, 8, 9, 10}.

Find

i) n(P)     ii) n(Q)

Solution :

P = {1, 5, 7, 8, 10} and Q = {1, 4, 5, 8, 9, 10}.

Number of elements in the set P is 5

n(P) = 5

Number of elements in set Q is 6

n(Q) = 6

Problem 10 :

List all the subsets of {w, x, y, z}

Solution :

{w, x, y, z}

Subsets are,

{ }, {w}, {x}, {y}, {z}, {w, x}, {w, y}, {w, z}, {x, y}, {x, z}, {y, z}, {w, x, y}, {w, y, z}, {w, z, x}, {x, y, z}, {w, x, y, z}

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