﻿ ﻿Setting Things Up with Basic Set Theory - Picturing and Measuring - Graphs, Measures, Stats, and Sets - Basic Math & Pre-Algebra For Dummies

## Basic Math & Pre-Algebra For Dummies, 2nd Edition (2014)

### Chapter 20. Setting Things Up with Basic Set Theory

In This Chapter Defining a set and its elements Understanding subsets and the empty set Knowing the basic operations on sets, including union and intersection

A set is just a collection of things. But in their simplicity, sets are profound. At the deepest level, set theory is the foundation for everything in math.

Set theory provides a way to talk about collections of numbers, such as even numbers, prime numbers, or counting numbers, with ease and clarity. It also gives rules for performing calculations on sets that become useful in higher math. For these reasons, set theory becomes more important the higher up you go the math food chain — especially when you begin writing mathematical proofs. Studying sets can also be a nice break from the usual math stuff you work with.

In this chapter, I show you the basics of set theory. First, I show you how to define sets and their elements and how you can tell when two sets are equal. I also show you the simple idea of a set's cardinality. Next, I discuss subsets and the all-important empty set (∅). After that, I discuss four operations on sets: union, intersection, relative complement, and complement.

Understanding Sets

A set is a collection of things, in any order. These things can be buildings, earmuffs, lightning bugs, numbers, qualities of historical figures, names you call your little brother, whatever. You can define a set in a few main ways:

· Placing a list of the elements of the set in braces { }: You can simply list everything that belongs in the set. When the set is too large, you use an ellipsis (...) to indicate elements of the set not mentioned. For example, to list the set of numbers from 1 to 100, you can write {1, 2, 3, ..., 100}. To list the set of all the counting numbers, you can write {1, 2, 3, ...}.

· Using a verbal description: If you use a verbal description of what the set includes, make sure the description is clear and unambiguous so you know exactly what's in the set and what isn't. For instance, the set of the four seasons is pretty clear-cut, but you may run into some debate on the set of words that describe my cooking skills because different people have different opinions.

· Writing a mathematical rule (set-builder notation): In later algebra, you can write an equation that tells people how to calculate the numbers that are part of a set. Check out Algebra II For Dummies, by Mary Jane Sterling (Wiley), for details.

Sets are usually identified with capital letters to keep them distinct from variables in algebra, which are usually small letters. (Chapter 21 talks about using variables.)

The best way to understand sets is to begin working with them. For example, here I define three sets:

· A = {Empire State Building, Eiffel Tower, Roman Colosseum}

· B = {Albert Einstein's intelligence, Marilyn Monroe's talent, Joe DiMaggio's athletic ability, Sen. Joseph McCarthy's ruthlessness}

· C = the four seasons of the year

Set A contains three tangible objects: famous works of architecture. Set B contains four intangible objects: attributes of famous people. And set C also contains intangible objects: the four seasons. Set theory allows you to work with either tangible or intangible objects, provided that you define your set properly. In the following sections, I show you the basics of set theory.

Elementary, my dear: Considering what's inside sets

The things contained in a set are called elements (also known as members). Consider the first two sets I define in the section intro:

· A = {Empire State Building, Eiffel Tower, Roman Colosseum}

· B = {Albert Einstein's intelligence, Marilyn Monroe's talent, Joe DiMaggio's athletic ability, Sen. Joseph McCarthy's ruthlessness}

The Eiffel Tower is an element of A, and Marilyn Monroe's talent is an element of B. You can write these statements using the symbol ∈, which means “is an element of”: However, the Eiffel Tower is not an element of B. You can write this statement using the symbol ∉, which means “is not an element of”: These two symbols become more common as you move higher in your study of math. The following sections discuss what's inside those braces and how some sets relate to each other.

Cardinality of sets

The cardinality of a set is just a fancy word for the number of elements in that set.

When A is {Empire State Building, Eiffel Tower, Roman Colosseum}, it has three elements, so the cardinality of A is three. Set B, which is {Albert Einstein's intelligence, Marilyn Monroe's talent, Joe DiMaggio's athletic ability, Sen. Joseph McCarthy's ruthlessness}, has four elements, so the cardinality of B is four.

Equal sets If two sets list or describe the exact same elements, the sets are equal (you can also say they're identical or equivalent). The order of elements in the sets doesn't matter. Similarly, an element may appear twice in one set, but only the distinct elements need to match.

Suppose I define some sets as follows:

· C = the four seasons of the year

· D = {spring, summer, fall, winter}

· E = {fall, spring, summer, winter}

· F = {summer, summer, summer, spring, fall, winter, winter, summer}

Set C gives a clear rule describing a set. Set D explicitly lists the four elements in C. Set E lists the four seasons in a different order. And set F lists the four seasons with some repetition. Thus, all four sets are equal. As with numbers, you can use the equals sign to show that sets are equal:

· C = D = E = F

Subsets

When all the elements of one set are completely contained in a second set, the first set is a subset of the second. For example, consider these sets:

· C = {spring, summer, fall, winter}

· G = {spring, summer, fall}

As you can see, every element of G is also an element of C, so G is a subset of C. The symbol for subset is ⊂, so you can write the following:

· G ⊂ C Every set is a subset of itself. This idea may seem odd until you realize that all the elements of any set are obviously contained in that set.

Empty sets

The empty set — also called the null set — is a set that has no elements:

· H = {}

As you can see, I define H by listing its elements, but I haven't listed any, so H is empty. The symbol ∅ is used to represent the empty set. So H = ∅.

You can also define an empty set by using a rule. For example,

· I = types of roosters that lay eggs

Clearly, roosters are male and, therefore, can't lay eggs, so this set is empty. You can think of ∅ as nothing. And because nothing is always nothing, there's only one empty set. All empty sets are equal to each other, so in this case, H = I.

Furthermore, ∅ is a subset of every other set (the preceding section discusses subsets), so the following statements are true:

· ∅ ⊂ A

· ∅ ⊂ B

· ∅ ⊂ C

This concept makes sense when you think about it. Remember that ∅ has no elements, so technically, every element in ∅ is in every other set.

Sets of numbers

One important use of sets is to define sets of numbers. As with all other sets, you can do so either by listing the elements or by verbally describing a rule that clearly tells you what's included in the set and what isn't. For example, consider the following sets: My definitions of J and K list their elements explicitly. Because K is infinitely large, you need to use an ellipsis (...) to show that this set goes on forever. The definition of L is a description of the set in words.

I discuss some especially significant sets of numbers in Chapter 25.

Performing Operations on Sets

In arithmetic, the Big Four operations (adding, subtracting, multiplying, and dividing) allow you to combine numbers in various ways (see Chapters 3 and 4 for more information). Set theory also has four important operations: union, intersection, relative complement, and complement. You'll see more of these operations as you move on in your study of math.

Here are definitions for three sets of numbers: In this section, I use these three sets and a few others to discuss the four set operations and show you how they work. (Note: Within equations, I relist the elements, replacing the names of the sets with their equivalent in braces. Therefore, you don't have to flip back and forth to look up what each set contains.)

Union: Combined elements

The union of two sets is the set of their combined elements. For example, the union of {1, 2} and {3, 4} is {1, 2, 3, 4}. The symbol for this operation is ∪, so Similarly, here's how to find the union of P and Q: When two sets have one or more elements in common, these elements appear only once in their union set. For example, consider the union of Q and R. In this case, the elements 4 and 6 are in both sets, but each of these numbers appears once in their union: The union of any set with itself is itself: Similarly, the union of any set with ∅ (see the earlier “Empty sets” section) is itself: Intersection: Elements in common

The intersection of two sets is the set of their common elements (the elements that appear in both sets). For example, the intersection of {1, 2, 3} and {2, 3, 4} is {2, 3}. The symbol for this operation is ∩. You can write the following: Similarly, here's how to write the intersection of Q and R: When two sets have no elements in common, their intersection is the empty set (∅ ): The intersection of any set with itself is itself: But the intersection of any set with ∅ is ∅: Relative complement: Subtraction (sorta)

The relative complement of two sets is an operation similar to subtraction. The symbol for this operation is the minus sign (–). Starting with the first set, you remove every element that appears in the second set to arrive at their relative complement. For example,

· {1, 2, 3, 4, 5} – {1, 2, 5} = {3, 4}

Similarly, here's how to find the relative complement of R and Q. Both sets share a 4 and a 6, so you have to remove those elements from R:

· R – Q = {2, 4, 6, 8, 10} – {4, 5, 6} = {2, 8, 10}

Note that the reversal of this operation gives you a different result. This time, you remove the shared 4 and 6 from Q:

· Q – R = {4, 5, 6} – {2, 4, 6, 8, 10} = {5} Like subtraction in arithmetic, the relative complement is not a commutative operation. In other words, order is important. (See Chapter 4 for more on commutative and non-commutative operations.)

Complement: Feeling left out

The complement of a set is everything that isn't in that set. Because “everything” is a difficult concept to work with, you first have to define what you mean by “everything” as the universal set (U). For example, suppose you define the universal set like this:

· U = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9}

Now, here are a couple of sets to work with: The complement of each set is the set of every element in U that isn't in the original set: The complement is closely related to the relative complement (see the preceding section). Both operations are similar to subtraction. The main difference is that the complement is always subtraction of a set from U, but the relative complement is subtraction of a set from any other set.

The symbol for the complement is ′, so you can write the following: ﻿