Power Set

Definition

The power set of a set is the set of all its subsets.

Example:

A = {1, 2}

Subsets of A:

{}
{1}
{2}
{1, 2}

Power set:

{{}, {1}, {2}, {1, 2}}

Power set = collection of every possible subset.

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Empty Set Is Included

The empty set is a subset of every set.

Example:

A = {1, 2}

The empty set:

{}

is included in the power set of A.

The power set always contains the empty set.

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The Original Set Is Included

Every set is a subset of itself.

So the original set also appears inside its power set.

Example:

A = {1, 2}

The subset:

{1, 2}

is included in the power set.

The power set always contains the original set.

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Power Set Grows Quickly

If a set has n elements, its power set has:

2^n

subsets.

Example:

A = {1, 2, 3}

has 3 elements.

Its power set has:

2^3 = 8

subsets.

Power sets grow by possible inclusion/exclusion choices.

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Final Summary

• The power set is the set of all subsets.
• It includes the empty set.
• It includes the original set.
• A set with n elements has 2^n subsets.
• Power sets matter because they describe all possible selections from a set.