Basic Math and Pre-Algebra For Dummies (93 page)

The intersection of any set with itself is itself:

But the intersection of any set with ∅ is ∅:

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.)

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:

Part V

 For more info on simplifying algebraic expressions and to discover a trick that makes factoring quadratic polynomials easy, go to
www.dummies.com/extras/basicmathandprealgebra
.

In this part…

• Evaluate, simplify, and factor algebraic expressions
  
• Keep algebraic equations balanced and solve them by isolating the variable
  
• Use algebra to solve word problems too difficult to solve with just arithmetic
  

Chapter 21

In This Chapter

Meeting Mr. X head-on

Understanding how a variable such as
x
stands for a number

Using substitution to evaluate an algebraic expression

Identifying and rearranging the terms in any algebraic expression

Simplifying algebraic expressions

You never forget your first love, your first car, or your first
x.
Unfortunately for some folks, remembering their first
x
in algebra is similar to remembering their first love who stood them up at the prom or their first car that broke down someplace in Mexico.

The most well-known fact about algebra is that it uses letters — like
x
— to represent numbers. So if you have a traumatic
x
-related tale, all I can say is that the future will be brighter than the past.

What good is algebra? That question is a common one, and it deserves a decent answer. Algebra is used for solving problems that are just too difficult for ordinary arithmetic. And because number crunching is so much a part of the modern world, algebra is everywhere (even if you don't see it): architecture, engineering, medicine, statistics, computers, business, chemistry, physics, biology, and, of course, higher math. Anywhere that numbers are useful, algebra is there. That fact is why virtually every college and university insists that you leave (or enter) with at least a passing familiarity with algebra.

In this chapter, I introduce (or reintroduce) you to that elusive little fellow, Mr.
X,
in a way that's bound to make him seem a little friendlier. Then I show you how
algebraic expressions
are similar to and different from the arithmetic expressions that you're used to working with. (For a refresher on arithmetic expressions, see Chapter
5
.)

In math,
x
stands for a number — any number. Any letter that you use to stand for a number is a
variable,
which means that its value can
vary
— that is, its value is uncertain. In contrast, a number in algebra is often called a
constant
because its value is
fixed.

Sometimes you have enough information to find out the identity of
x.
For example, consider the following:

• 2 + 2 =
  x
  

Obviously, in this equation,
x
stands for the number 4. But other times, what the number
x
stands for stays shrouded in mystery. For example:

• x
  > 5
  

In this inequality,
x
stands for some number greater than 5 — maybe 6, maybe
, maybe 542.002.

In Chapter
5
, I introduce you to arithmetic expressions: strings of numbers and operators that can be evaluated or placed on one side of an equation. For example:

In this chapter, I introduce you to another type of mathematical expression: the algebraic expression. An
algebraic expression
is any string of mathematical symbols that can be placed on one side of an equation and that includes at least one variable.

Here are a few examples of algebraic expressions:

 
As you can see, the difference between arithmetic and algebraic expressions is simply that an algebraic expression includes at least one variable.

In this section, I show you how to work with algebraic expressions. First, I demonstrate how to evaluate an algebraic expression by substituting the values of its variables. Then I show you how to separate an algebraic expression into one or more terms, and I walk through how to identify the coefficient and the variable part of each term.

 To evaluate an algebraic expression, you need to know the numerical value of every variable. For each variable in the expression, substitute, or plug in, the number that it stands for and then evaluate the expression.

In Chapter
5
, I show you how to evaluate an arithmetic expression. Briefly, this means finding the value of that expression as a single number (flip to Chapter
5
for more on evaluating).