Algebra/Chapter 2/Sets

Algebra

2.6: Sets

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In this section we mainly set up some useful notation. While the ideas here are not very central to the study of algebra, they do come up from time to time, so pay attention! It is exactly because these ideas don't reoccur on every page that they can be confusing when they suddenly come up later on. So be prepared to revisit this section as necessary to refresh your memory.

Sets and the Number Line

A set is a collection of things. They are also an example of an expression, as sets also describe mathematical objects of interest, in this case a group of objects.

Examples of sets might be the collection of letters used in the English alphabet, or the set of books written by John Steinbeck. For us, the sets we will discuss will usually be collections of numbers because these are the sets that are important in algebra. Each of the things in a set is called an element of the set. The number of elements in a set could be finite or could be infinite. The only requirement is that the elements of a set should be described explicitly in some way either now or in the future (after we solve some problem). In this book we will mostly try to use capital letters as the symbols for sets whereas lower case letters are often (but not always!) used for variables.

There may be no elements in a set; such a set is called an empty set or a null set. For the empty set, we use the symbol ∅ to represent it.

A set can be written by putting braces, that is { and }, around a list of the elements of the set, with each element being separated by a comma. For example, a set S containing natural or whole numbers from 1 to 10 could be shown as follows:

$S=\{1,2,3,4,5,6,7,8,9,10\}\,$

It is not always possible to list out all the numbers in a set. In these cases we rely on English to describe the set. That's right! Words are an important part of math too. The last and probably most common notation involves using variables and algebraic expressions, together with a description of what values the variables may take. For example, to describe the set of numbers that are a perfect square, we might write:

$\{n^{2}\mid n{\text{ is a whole number.}}\}$

or we can even use other sets in the description like:

${\Big \{}n^{2}\mid n\in \{0,1,2,3,\ldots \}{\Big \}}$

Sometimes we want to explicitly make clear that a particular number (or thing) is in a set. Keeping S from the example above, we know that 2 is an element of S, rather than writing this out in English, sometimes people use the shorthand 2 ∈ S. The symbol ∈ is chosen because it looks like an E, and E is the first letter of the word "element". If we want to express that something is not part of a set we use the symbol ∉. So to continue our example we know that 11 ∉ S.

Let's take a look at a practice problem.

Problem. For each of the following sets decide if 7 is in the set:

1. A={1, 2, 3, 5}
2. The set O of odd numbers.
3. C={1, 2, 3, 4, 5, …, 100}
4. The set P of all prime numbers.

Answer.

1. We can see by inspection that 7 ∉ A.
2. 7 ∈ O because 7 is an odd number.
3. 7 ∈ C. This is a bit of a trick question. While it hasn't been explicitly said it yet, when listing elements people sometimes use … to mean keep "following the pattern". And in the rest of mathematics … keeps getting used in this way, so it is time to get used to it. Here the pattern is we start with 1 and keep making the numbers 1 bigger, so yes, if we keep going we will reach 7.
4. 7 ∈ P. This problem requires that we try to factor 7, or have previously known it was a prime number. Since 2, 3, 4, 5, 6 all leave a remainder when we divide 7 by them, we see that 7 is a prime number.

We now introduce the basic ideas that come up when you have two or more sets.

Subsets and Super sets

Sometimes every element in one set is contained in another set. For example, let S = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10} and T = {2, 4, 6, 8, 10}. Clearly T is just the even numbers between 1 and 10, and every number in T is already in S. In this example we would say that T is a subset of S. Instead of saying which set is the smaller one, we could instead say which set is the bigger one by calling it a superset. That is, we could say S is a superset of T. As one deals with sets more and more, it becomes increasingly tempting to say that T is smaller than S (or maybe less than S). In fact, we already have! Because the relationship of one set being contained in another is so similar to relationship of one number being less than another it is natural to introduce a symbol very similar to inequality symbol <. To avoid confusing sets with numbers we don't want to use exactly the same symbol, so we will round out the point a bit and use the symbol ⊂. So instead of writing out in words "T is a subset of S", we could write T ⊆ S. Just like inequality we can flip the symbol around. We just have to make sure the rounded side points to the smaller thing. That is, for our example we could write S ⊃ T.

What if two sets have exactly the same elements? In this case we say that the two sets are equal. So if someone else came along and said let U = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}, then we say that S and U are equal. This time we don't worry about confusing sets and numbers and we will stick with the symbol = to denote when two sets are equal. So we can write S = T. Are there relationships corresponding to ≤ and ≥? Yes, the are ⊆ and ⊇ and they work like you might expect. Here is a table that explains how each of these symbols work.

The empty set is a subset of all sets.

Statement	Meaning
T ⊂ S	Means T is a subset of S. That is every element of T is an element in S, but there is some element of S that is not in T.
T ⊃ S	Means S is a subset of T. That is, every element of S is an element in T, but there is some element of T that is not in S.
S = T	Means that S and T have exactly the same elements. That is, use a few more words every element of S is an element of T and every element of T is an element of S.
T ⊆ S	Means every element of T is an element of S. (Notice that saying it this way allows for the possibility that S = T)
S ⊇ T	Means every element of T is an element of S.

Venn Diagrams

A Venn Diagram shows the logical relation between sets using simple closed curves such as circles drawn a plane.

John Venn

Unions and Intersections

There are two other things to do with sets. Given sets in S and T we may want to talk about all the elements that are in either S or T. Since a set is just a collection of things, and "the elements that are in either S or T" is a collection, you can think of this as defining a new set called the union of S and T. We write the union of S and T by S ∪ T. We use the symbol ∪ because it looks like a u, and u is the first letter in word "union". Let's do an example.

Problem. Suppose S = {2, 4, 6, 8, 10} and T = {3, 6, 9, 12, 15}, Find S ∪ T.

Answer. S ∪ T = {2, 3, 4, 6, 8, 9, 10, 12, 15}

An important thing to notice in this example is that S ∪ T doesn't contain two 6's. The union contains all the elements in either set, but 6 is still just one thing that happens to be in both sets.

Give sets S and T, instead of thinking about things that are in either set, it is sometimes handy to think about things that are in both sets. Again we think of the collection of "the elements in both S and T, this set is called the intersection of S and T. We write the intersection of S and T by S ∩ T. We don't use the symbol ∩ because it looks like an i. It doesn't. Somehow, an i between two symbols just wouldn't look as good, so we want to pick something else. This symbol is just the upside down symbol of the symbol for union. Let's consider what the intersection looks like in the problem above.

Problem. Suppose S = {2, 4, 6, 8, 10} and T = {3, 6, 9, 12, 15}, Find S ∩ T.

Answer. S ∩ T = {6}

The number 6 is the only number in both sets, so it is the only element of the intersection.

Subscripts

In a sentence like "let x and y be numbers", we can presume the possibility that $x=y$. If we wish to exclude this possibility, we state it explicitly. We say "let x and y be distinct numbers". We can keep going on with this pattern, talking about more than two variables, for example in the sentence "let x, y, and z be numbers". But there are only twenty-six letters in the alphabet. With such limited options to choose from, we would obviously run out of letters in enumerating variables. Hence we use a new notation with subscripts, tiny numbers written to the side of a number. We can use this new notation as follows:

$Let\ x_{1},\ x_{2}\ be\ numbers.$ $Let\ x_{1},\ x_{2},\ x_{3}\ be\ numbers.$ $Let\ x_{1},\ x_{2},\ x_{3},\ x_{4}\ be\ numbers.$

The most general case of this sentence is of course:

$Let\ x_{1},......,x_{n}\ be\ numbers.$

Where n represents the amount of objects in a set.