Recognizing Power Sets in Algebra

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  • 0:01 Introduction to Sets
  • 1:20 Visual Example of a Power Set
  • 3:29 Numerical Example of a…
  • 6:16 Recognizing Power Sets
  • 9:16 Lesson Summary
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Lesson Transcript
Instructor: Maria Airth

Maria has a Doctorate of Education and over 20 years of experience teaching psychology and math related courses at the university level.

This lesson will give students a very brief introduction to sets and subsets as a lead into how to determine if a set is actually a power set. Through the use of examples and trials, students will learn to identify true power sets.

Introduction to Sets

Hi, and welcome to this lesson on Recognizing Power Sets in Algebra.

I suppose the best place to start is to discuss what a set is. A set is a gathering together (or a collection) of certain items or elements. That's all; just bringing things together. You can have sets of many things. You can have sets of toys, fruit, animals, or anything at all really.

Think of it like a collection. Do you collect movie series? Well, to have a set of any series, you would have to have all the movies from the series. That makes the set. If you did not have all of the movies in a series, you would have what is called a subset. A subset is a set formed from the elements of another set. In the movie example, if you had the first two movies from the Lord of the Rings trilogy but not the third, you would have a subset of the movie series set. Maybe you have the first movie and the third, but not the second - that is another subset of the movie series whole set.

Visual Example of a Power Set

Let's look at a visual to see just how many subsets you can get out of a set. I will use ice cream for this example: strawberry, chocolate, and vanilla. Written algebraically using initials, the set would look like this: {S, C, V}. So, how many subsets can we get out of this three-element set? The question can really be posed as: How many different ways can these three items be arranged?

First, you have a subset of nothing at all, or { }. Yes, an empty set is a subset of every set.

Then you have a subset consisting of each flavor by itself: {S}, {C}, and {V}. The next set of subsets would be the different pairings of two choices each, such as: {S,C}, {S,V}, and {C,V}. The final subset is the subset that includes all the elements: {S,C,V} These eight options are all of the possible subsets for the set {S,C,V}.

Together, all of these subsets make a power set. A power set is a set that contains all of the possible subsets of a given set. The power set is noted like this: P(S) = . So, for our ice cream example, the power set notation would be:

P(S,C,V) = {{ }, {S}, {C}, {V}, {S,C}, {S,V}, {C,V}, {S,C,V}}

Did you notice that the sets of none of the elements and all of elements are subsets as well as every combination of the elements? That is important to remember.

Numerical Example of a Power Set

Why would you ever need to use power sets in algebra (or in real life, for that matter)?

Well, power sets can be used to identify all the different ways items can be put together. Maybe you need to work out the different possible groups for a set of students, or maybe you want to figure out how to place a collection of items in a shop window; you can use power sets to identify all the different ways you could group your items together.

But another use for them is determining all the factors of a number. Power sets are an amazing tool to find every single factor for any number imaginable.

Okay, so, can you tell me all of the factors of the number 130? (Remember that a factor is a number that can be multiplied by another number to result in the given product.)

To determine all of the factors for 130, we first start by identifying the prime factors, which are those factors of a given number that can only be divided by themselves and 1.

Set of Prime Factors for 130 is (S) = {2, 5, 13}

Now, we create a table to display the power set of this set of prime factors.

The first row in the table is the empty set, and a factor of 1 is assigned because 1 is the multiplicative identity. The next three rows are the subsets with a single element; the actual element will be the resulting factor. The next three rows are the subsets with just two elements. The factor listed is the product of the two elements in the subset. Finally, you have the subset that consists of all the elements from the original set and the factor is, of course, the given number from the example.

Subset Factor
1 { } 1
2 {2} 2
3 {5} 5
4 {13} 13
5 {2,5} 10
6 {2,13} 26
7 {5,13} 65
8 {2,5,13} 130

So, the factors of 130 are: 1, 2, 5, 13, 10, 26, 65, and 130. And this was found by using power set notation!

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