In exercises above, you grouped the elements or members of the set into different subsets.

In this case, the elements of the set are the 25 different coins.

A subset is also a set. When we talk about subsets, it is in reference to another set. For example, the set of pennies in the collection above is a subset of the collection of coins.

To show that one set is not a subset of another set, there would have to be an element in the first set that is not in the second set.

There is one thing about sets that confuses a lot of people. By definition, we say that a set is a collection of objects. But in reality, a set can have no elements in it.

Look at the definition of a subset very carefully. Is the null set is a subset of every set?

I hope this was your conclusion: The null set is a subset of every set.

Note that in the definition of a subset above, I used the letters B and C for the sets. We do not have to use these letters. They are just dummy variables much like when we use x, y , and n in algebra!! Here is another perfectly fine way to define subset: A set R is a subset of a set Q if every element of set R is an element of set Q.

We usually use a lot of abbreviations in math so that it isn’t so cumbersome writing everything out. Many times, we pick a capital letter to represent a set or subset and define the set by that letter. I will define some sets now: The set of 25 coins we’ve been working with will be defined as C. The subsets of pennies, nickels, dimes, quarters and half dollars will be denoted by P, N, D, Q and H, respectively. Notice I gave them meaningful letters so it is easy to remember which letter stands for which subset.

One way to describe a set is to list its objects, if possible. (Later, we will discuss why this is not always practical or possible.) When we list the objects, we do so inside of braces using commas to separate the elements. The order in which the elements are listed is irrelevant. Using the listing method, one way to write our coin collection, set C, is

C = { 6P, 7N, 6Q, 7P, 5P, 6H, 7Q, 5D, 7D, 8Q, 4D, 6N, 5N, 7H, 6D, 4N, 9Q, 9H, 5H, 8P, 8D, 4Q, 5Q, 9D, 8N }

The way to express that an element, like 4D, is in set C is with the symbol which means “is an element of”. We write: 4D (read “4D is an element of C“).

Let A represent the one dollar coins in our set C. Notice that set A is empty. We can show that A contains no elements by writing A = { }. The null or empty set is also written Ø. So, we can also write A = Ø. In fact, { } = Ø always. It is correct to write the null or empty set either way. But note that it is not correct to write the null set like this: {Ø}. That would be a set that does contain at least one element the null set! See how this can be a bit confusing?

The symbol ⊆ means “is a subset of” and so the way to express that P is a subset of C is by writing . Since N is also a subset of C, we can write N ⊆ C. If the first set has fewer elements than the second set, then the first set is a proper subset of the second set. In this case, the symbol ⊂ may be used in place of ⊆. So, in the case of the coins, it is also correct to write P ⊂ C and N ⊂ C. A first set is not a subset of a second set if there is at least one element in the first set that is not in the second set. Think again about why the null set has to be a subset of every set. For it not to be a subset, there would have to be at least one element in it that wasn’t in the second set. But, the null set is empty!

Something else to consider when working with sets is how many elements are in a given set.

Think of the n as the number of objects in the set. Do you remember any functional notation from algebra like f(x) = 2x + 3 implies f(5) = 2(5) + 3 = 10 + 3 = 13 – that is, when you plug 5 into the function f(x), out pops 13? Well, when working with the cardinality of a set, it is the same kind of thing and we use similar notation. In this case, when you plug C into the function n, out pops the number 25!!! When you are asked about the cardinality of a set, the answer is a number. We say that two sets are equivalent if they each have the same number of elements which is to say they have the same cardinality. We use the symbol, ~, to denote equivalence. For instance, since P and H each contain four elements, we can write P ~ H. Two sets are equal if they have the exact same elements.

What if you were asked which coins were pennies and also from the year 1965? In set C, there is only one coin that satisfies both criteria and that is 5P. When trying to find the elements that are common to two sets, in this case, P (pennies) and V (coins from 1965), we use the word intersection. A formal definition of intersection follows:

Using the correct notation, we could now write P ∩ V = {5P}

If we look at the two coin subsets D and Q, we see that their intersection is the empty set (D ∩ Q = Ø). Can you name three other pairs of sets that have an intersection that is the empty set?

What if you were asked which coins were either pennies or from the year 1965? List which coins would satisfy one or both of these conditions:

Did you get eight different coins this time? ___________________ Any coin having a 5 or P or both would do it. You just found the union of the two sets P and V. A formal definition of union follows:

Using correct notation, we write P ∪ V = {6P, 7P, 5P, 5D, 5N, 5H, 8P, 5Q}. Remember that the order in which the elements are written is irrelevant! IMPORTANT: If an element is in both sets, it is only written down once in the union.

Put aside the coin set for now. Let’s work with another set. When working with sets, we need to be clear about what we are talking about. Here is another definition for you.

Did you notice that the order matters for difference, but not for union or intersection?

Make sure you can work through all of the problems above perfectly on your own. If you have trouble understanding any of them, check with your classmates or ask your teacher if something still needs clarifying!!! Then, make up some more problems of your own as instructed below before going on.

Remember when we mentioned that the listing method wasn’t always practical or possible? Consider a universal set that contained all of the whole numbers. These could not be listed. Another way of expressing this set would be by description. You could say U = (the whole numbers) This is a very formal way of writing this same set U = {x|x is a whole number} which is read “U is the set of all x such that x is a whole number”. You may have seen this notation in algebra. We won’t be so formal here. Another way to get around the listing problem is to use three consecutive dots which means “and so on”. We would write U = {0,1,2,3,…}. Here is a way to express all the even numbers between 13 and 509: {14,16,18,…,508}. You could list them out (if you didn’t have a life!), but you can see the advantage of using three dots!!!! Using the three dots doesn’t always work either. Consider the set of all real numbers. They cannot be listed. You would have to write something like this: {all real numbers) or {x|x is a real number}.

It’s now time get out your A-blocks. They can be found on Material Cards 2A – 2E. The white cards are value label cards (which you won’t use until Exercise Set 3), and the colored objects are the actual A-blocks. There are 24 objects in each set of A-blocks.

We will use the following abbreviations when referring to each of the 24 elements.

We’ll use three letters to specify each object in this order – size, color, shape. The small blue square would be denoted by SBQ and the large red circle would be denoted by LRC. These abbreviations are on the objects that you cut out.

Let’s define our universe of A-blocks by A.

Put your A-blocks away for now. Don’t lose them!! Before we continue working with the A-blocks, we’ll need to learn about and work with Venn diagrams. We will then come back and do more work with our A-blocks.

Let’s consider a situation where a mother, Mary, has three children –- Alicia, Brent and Carlos. Mary has red hair. The question we will attempt to answer is this: What possibilities exist regarding which, if any of her children, have red hair? List the possibilities you came up with.

Let’s work through the above problem. First of all, let’s break the problem down a little bit. It’s possible none of her children have red hair, only one has red hair, two out of three have red hair or all three of them ended up as redheads. Within each of these four situations, let’s write down each of the possibilities. We will list each possibility as a set, using the childrens’ names as the possible elements.

We came up with eight different possibilities.

The above problem really amounts to a set theory question. Given a set M, where M = {A, B ,C}, list all possible subsets of M. First of all, remember that the null set is a subset of every set and the entire set is a subset of itself (it’s just not a proper subset). You did remember those two points, didn’t you?

Let’s begin by listing the possibilities according to how many elements might be in the subsets either no elements, one element, two elements or three elements. For each of these situations, list all possibilities:

Here is a list of all possible subsets: { }, {A}, {B}, {C}, {A, B}, {A, C}, {B, C}, {A, B, C}

WOW! Did you see how we came up with the same eight possibilities? This set theory stuff is just amazing, isn’t it?

There is one more operation we will define on sets. So far, you’ve worked with union, intersection, difference and complement. Now we will learn what it means to take the Cartesian product of two sets, A and B.

In general, for any two sets, F and G, n(F × G) = n(G × F).

Let’s try a more challenging problem: Find {(4, 3), 5) × {(3, 3),{4, 7, 5, 2}, 1}