We can also have expressions that represent values which are either true or false. These expressions are called Boolean expressions. Usually such expressions involve the comparison of values with the following operators: > (greater than), < (less than), and = (equals). For example, consider the following three Boolean expressions:

 (6 > 10) (6 > 6) (6 = 10) (6 < 10)
 false false false true

Languages such as C++ that use the = for the assignment operation often use == (double equals) to test for equality. Be careful not to confuse the two when programming in these languages.

We can also have some boolean expressions with two operators that compare relative size and equality. These two operators are >= (greater than or equal to) and <= (less than or equal to).

 (6 >= 10) (6 >= 6) (6 <= 10)
 false true true

Since these are Boolean expressions, each statement is either true or false. Of course, the answers are quite easy to determine because we know the relationship between 6 and 10. However, Boolean expressions can also be constructed from variables like the following one:

 (count < 100)
 ???

Without knowing the value of the variable count we cannot determine the value of the Boolean expression. Quite often an expression like this is used to control a repetitive action. Each time the action is performed, the value of count increases until the expression eventually becomes false. We will see more about this kind of control in a few more lessons.

Boolean expressions can be joined together to form longer expressions using three logical operators: AND, OR, and NOT. Suppose we want to test our count variable to determine if the count is between 10 and 100. We can write a Boolean expression using the AND operator to construct our test:

 (count > 10) AND (count < 100)

In order for our entire Boolean expression to be true, both of the smaller expressions must be true. If count is equal to 1000, then only the first half of the expression is true, and the entire expression evaluates to false. This is exactly what we want because 1000 is not between 10 and 100. If count is equal to 1, then only the second half of the expression is true, and the entire expression evaluates to false. This is also what we want since 1 is also not between 10 and 100.

Now let's consider how we could write a Boolean expression to test whether count is outside the range 10 to 100. One easy solution would be to reverse the test we have already created using the NOT operator. This operator turns true values to false and false values to true. Therefore, if we want to construct the opposite test, we can reverse the test we already have like this:

 NOT ((count > 10) AND (count < 100))

Although constructing this new test only requires a small change to our previous test, the meaning of the new test is not very clear. A better way to construct this test is to use the OR operator. What we really want to test is when count is greater than 100 or less than 10.

 (count < 10) OR (count > 100)

In order for our entire Boolean expression to be true, at least one of the smaller expressions must be true. If count is equal to 1000, then the second half of the expression is true, and our entire test is also true. This is exactly what we want because 1000 is not between 10 and 100. If count is equal to 1, then the first half of the expression is true, and the entire test is also true. This is also what we want since 1 is not between 10 and 100.

Often the truth values for the Boolean operators AND, OR, and NOT are described in a table called a truth table. The table lists all the possible operands for a given operator and the resulting truth values. Truth tables for AND, OR, and NOT are listed below. The values in parentheses represent a Boolean expression that is either true (T) or false (F).

 AND operator
 Boolean Expression Truth Value (T) AND (T) T (T) AND (F) F (F) AND (T) F (F) AND (F) F

 OR operator
 Boolean Expression Truth Value (T) OR (T) T (T) OR (F) T (F) OR (T) T (F) OR (F) F

 NOT operator
 Boolean Expression Truth Value NOT (T) F NOT (F) T

To summarize, the AND operation is only true when it joins two true Boolean expressions. The OR operation is only false when it joins two false expressions. The NOT operation always reverses the truth value of the expression.