Basic Math and Pre-Algebra For Dummies (91 page)

Suppose you want to find the probability that six tossed coins will all fall heads up. To do this, you want to build a fraction, and you already know that the denominator — the number of total outcomes — is 64. Only one outcome is the target outcome, so the numerator is 1:

So the probability that six tossed coins will all fall heads up is
.

Here's a more subtle question: What's the probability that exactly five out of six tossed coins will all fall heads up? Again, you're building a fraction, and you already know that the denominator is 64. To find the numerator (target outcomes), think about it this way: If the first coin falls tails up, then all the rest must fall heads up. If the second coin falls tails up, then again all the rest must fall heads up. This is true of all six coins, so you have six target outcomes:

Therefore, the probability that exactly five out of six coins will fall heads up is
, which reduces to
(see Chapter
9
for more on reducing fractions).

Chapter 20

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.

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.

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.

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.

 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
  

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.