Basic Math and Pre-Algebra For Dummies (92 page)

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.

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
.

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.)

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:

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 (∅ ):