Power Set

The power set is a set that contains all of the subsets, as well as the empty set and the original set. It's usually symbolized with the letter P. The cardinality of a power set is determined by the number of subsets formed for a given set.  If set A = {x, y, z} is one set, then all of its subsets {x}, {y}, {z}, {x, y}, {y, z}, {x, z}, {x, y, z} and {} are the elements of power set, such as:

Power set of A, P(A) = { {x}, {y}, {z}, {x, y}, {y, z}, {x, z}, {x, y, z}, {} }

Where P(A) denotes the power set.

In this article, you will understand the power set, its subsets, and examples. 

Define Power Set with Example

Well! To define a power set, we would say it is simply a set of all the subsets of a set. Feeling confused? Let’s understand the power set with an example. Let’s say:

An ice cream parlor has strawberry, pineapple and butterscotch flavours

What Order do you Place?

• Absolutely Nothing: {}

• Or maybe just strawberry: {strawberry}. Or just {pineapple} or just {butterscotch}

• Or two together: {strawberry, pineapple} or {pineapple, butterscotch} or {butterscotch, strawberry}

• Or all three! {pineapple, butterscotch, strawberry}

Collectively, we get the Power Set of {x, y, and z}:

P(S) = {{}, {x}, {y}, {z}, {x, y}, {x, z}, {y, z}, {x, y, z}}

Subsets

The subset is a collection of sets or we can say all the different ways we can select the products. The order of the items being selected doesn’t matter, even selecting any, or all.

For the set with variables {x, y, and z}:

The empty set {} is a subset of {x, y, z}

These are subsets: {x}, {y} and {z}

These are also subsets: {x, y}, {x, z} and {y, z}

And {x, y, z} is a subset of {x, y, z}

Number of Subsets

It’s very simple! Let’s assume if the original set consists of ‘n’ members, then the Power Set will contain 2n members. This is to say:

{w, x, y, z} has 4 members (w,x,y and z).

Thus, the Power Set must have 24 = 16.

Properties of Power Set

The properties of the power set are:

• It's larger than the original set.

• The power set of A has 2ⁿ elements, where n is the number of elements in set A.

• The power set of any of the countable finite sets is considered as countable.

• We can make a one-to-one mapping of the resulting set, P(S), with the real numbers for a set of natural numbers.

• P(S) of set S denotes a Boolean Algebra example when used with the union of sets, the intersection of sets, and the complement of sets.

Cardinality of a Power Set

The total number of elements in a set is known as its cardinality. The list of all the subsets of a set is stored in a power set.  2n is the total number of subsets for a set of 'n' items. The cardinality of a power set is given by |P(A)| =  2n because the subsets of a set are the elements of a power set. The total number of elements in the given set is denoted by n.

Example: Set A = {1,2}; n = 2

|P(A)| = 2n = 22 = 4.

Subsets of A = {}, {1},{2},{1,2}

Therefore, |P(A)| = 4.

Power Set of Empty Set

When the number of elements in a set is 'n,' we know that the power set will have  2n elements. A set which has no elements is called an empty set. It is represented by {} or the symbol Ø. This means that {} is a subset of all sets. There are no elements in an empty set. As a result, the power set of the empty set is nothing but an empty set.

Because we just learned that an empty set has no elements, the power set of an empty set will have 20 elements. The power set of the empty set is an empty set with one element, i.e., 20 = 1. So, P(E) = {}.

Power Set of a Countable Set

We call a set countable when its element can be counted. Note that a countable set can be both finite and infinite.

For example, let’s take two sets

S1 = {B, C, D, F, G, H, J, K, L, M, N, P, Q, R, S, T, V, W, X, Z and Y} representing all consonants is a countably finite set.

On the other hand, S2 = {1, 2, 3, 4, 5, 6,……} depicting a set of natural numbers is a countably infinite set.

That said, the Power set of countably finite sets is always finite and thus countable.

For example, the set S1 representing consonants has 21 elements and its power set will have 221 = 2,097,152 elements. Hence, it is finite and therefore countable.

The power set of countably infinite sets is uncountable.

Power Set of an Uncountable Set

We call a set uncountable when its element is unable to be counted. Note that an uncountable set is always infinite. Remember that the Power set of an uncountable set is always uncountable.

For example, the set S3 depicting all fractional numbers between 1 and 10 is uncountable. Hence, the power set of uncountable sets is uncountable invariably.

Solved Examples

Solve the following question based on the power set.

Example 1: What will be the Cardinality of the Power Set of {0, 1, 2 . . ., 5}?

Solution:

The cardinality of a set is the number of elements that it consists of.

Given that, for a set S with 6 elements, its power set will be

n= 6

Now, the size of the power set is 26 = 64

Example 2: Find the power set of Z = {2, 7, 9} and a total number of elements.

Solution: 

Given, Z = {2, 7, 9}

Total number of elements in power set = 2n

Here, n = 3 (the number of elements present in set Z)

So,  23= 8, which shows that there are eight elements of the power set of Z

Therefore, 

P (Z) = [{}, {2}, {7}, {9}, {2, 7}, {7, 9}, {2, 9}, {2, 7, 9}]

Example 3: Find the number of elements that are present in the power sets of an empty set, set A = {}

Solution: 

In an empty set, the number of elements in a set is 'n'. We know that the power set will have 2n elements.

Since an empty set does not contain any elements, the power set will contain 20 elements or 1 element. Therefore, we can say that the power set of the empty set is an empty set, P(E) = {}.

Therefore, it is proved that the power set of the empty set is an empty set, P(E) = {}.

Example 4: In a food joint, we have our all-time favorite food: pizza, macaroni, sandwich, and burger. Evaluate the number of different ways can we have them?

Solution:

Let's use letters for the foods as: {P, M, S, B}. Selections include:

{} (nothing, you avoid fast food)

{P, M, S, B} (Every food item)

{M, S} (Macaroni and Sandwich are good together) etc

Let’s create a table using binary

And the arranged outcome is as follows:

P = { {}, {p},{m},{s},{b},{m,s}, {s,b}, {b,p}, {p,m}, }, {m,b}, {s,p}, {p,m,b}, {p,m,s}, [m,s,b}, {p,m,s,b}

Conclusion

This is all about the power set, its definition, and examples. Focus on the concept and solved examples to understand it well as this topic is used in different subjects apart from mathematics.