Power set facts for kids

The power set of a set is like a special collection of all the smaller groups you can make from that set. Imagine you have a set of toys. The power set would be a list of every single way you could pick those toys, including picking none of them, or picking all of them!

We write the power set of a set S as P(S).

If a set has n elements (meaning n items inside it), then its power set will always have 2ⁿ elements. This means the power set is always bigger than the original set, unless the original set is empty.

What is a Set?

A set is a collection of different things. These things are called elements or members of the set.

• For example, the set of primary colors is {red, yellow, blue}.
• The set of numbers {1, 2, 3} has three elements.

Sets are usually written with curly brackets `{}` around the elements. Each element in a set is unique, meaning it only appears once.

What is a Subset?

A subset is a set that contains only elements that are also found in another, larger set.

• If you have the set of fruits {apple, banana, orange}, then {apple, banana} is a subset of that set.
• The set {orange} is also a subset.
• Even { } (an empty set, meaning no fruits) is a subset.
• And the original set itself, {apple, banana, orange}, is also considered a subset of itself!

Every set has at least two subsets: the empty set (which has no elements) and the set itself.

How to Find a Power Set

Let's look at some examples to see how power sets work.

Example 1: A Set with Two Elements

Imagine you have a set with two elements, like the numbers {2, 5}. To find its power set, we need to list all possible subsets:

• The empty set: { } (This is always a subset!)
• Subsets with one element: {2}, {5}
• Subsets with two elements (the original set itself): {2, 5}

So, the power set of {2, 5} is: P({2, 5}) = { { }, {2}, {5}, {2, 5} }

Let's check the number of elements. The original set {2, 5} has 2 elements. Using the formula 2ⁿ, we get 2² = 4. Our power set has 4 elements, which matches!

Example 2: A Set with Three Elements

Now let's try a set with three elements, like {3, 4, 10}.

• The empty set: { }
• Subsets with one element: {3}, {4}, {10}
• Subsets with two elements: {3, 4}, {3, 10}, {4, 10}
• Subsets with three elements (the original set): {3, 4, 10}

So, the power set of {3, 4, 10} is: P({3, 4, 10}) = { { }, {3}, {4}, {10}, {3, 4}, {3, 10}, {4, 10}, {3, 4, 10} }

Let's check the number of elements. The original set {3, 4, 10} has 3 elements. Using the formula 2ⁿ, we get 2³ = 2 × 2 × 2 = 8. Our power set has 8 elements, which also matches!

Example 3: An Empty Set

What if the original set is empty? Let's say S = { }.

• The only subset of an empty set is the empty set itself.

So, the power set of { } is: P({ }) = { { } }

Using the formula 2ⁿ, the empty set has 0 elements (n=0). So, 2⁰ = 1. The power set has 1 element, which is correct!

Why is it called a Power Set?

The "power" in power set comes from the fact that the number of elements in the power set is always a power of 2 (2ⁿ). This mathematical idea is very useful in computer science, logic, and other areas of mathematics. It helps us understand all the possible combinations or choices we can make from a given group of items.