98 CHAPTER 3 Logic In this case, we normally interpret the word and to indicate that both events will take place. That is, the person must perform community service and must also pay a fine. Now suppose a judge states, “I sentence you to six months in prison or 10 months of community service.” In this case, we interpret the connective or as meaning the convicted person must either spend the time in jail or perform community service, but not both. The word or in this case is the exclusive or. When the exclusive or is used, one or the other of the events can take place, but not both. In a restaurant, a waiter asks, “May I interest you in a cup of soup or a sandwich?” This question offers three possibilities: You may order soup, you may order a sandwich, or you may order both soup and a sandwich. The or in this case is the inclusive or. When the inclusive or is used, one or the other, or both events can take place. In this chapter, when we use the word or in a logic statement, it will mean the inclusive or unless stated otherwise. If–then statements are often used to relate two ideas, as in the bank policy statement “If the average daily balance is greater than $500, then there will be no service charge.” If–then statements are also used to emphasize a point or add humor, as in the statement “If the Cubs win, then I will be a monkey’s uncle.” Now let’s look at logic from a mathematical point of view. Statements, Logical Connectives, and Quantifiers A sentence that can be judged either true or false is called a statement. Labeling a statement true or false is called assigning a truth value to the statement. Here are some examples of statements. 1. There are 100 inches in a yard. 2. The Empire State Building is in the state of New York. 3. The National Basketball Association (NBA) team from Portland, Oregon, is the Lakers. In each case, we may need to consult a resource such as the Internet, but we can determine that each statement is either true or false. Statement 1 is false because there are 36 inches in a yard, not 100 inches. Statement 2 is true because the Empire State Building is in New York City, which is in the state of New York. Statement 3 is false because the NBA team from Portland, Oregon, is the Trailblazers, not the Lakers. The Lakers are from Los Angeles, California. The three sentences discussed above are examples of simple statements because they convey one idea. Sentences combining two or more ideas that can be assigned a truth value are called compound statements. Compound statements are discussed shortly. Sometimes it is necessary to change a statement to its opposite meaning. To do so, we use the negation of a statement. For example, the negation of the statement “Emily is at home” is “Emily is not at home.” The negation of a true statement is always a false statement, and the negation of a false statement is always a true statement. We must use special caution when negating statements containing the words all, none (or no), and some. These words are referred to as quantifiers. Consider the statement “All lakes contain fresh water.” We know this statement is false because the Great Salt Lake in Utah contains salt water. Its negation must therefore be true. We may be tempted to write its negation as “No lake contains fresh water,” but this statement is also false because Lake Superior contains fresh water. Therefore, “No lakes contain fresh water” is not the negation of “All lakes contain fresh water.” The correct negation of “All lakes contain fresh water” is “Not all lakes contain fresh water” or “At least one lake does not contain fresh water” or “Some lakes do not contain fresh water.” These statements all imply that at least one lake does not contain fresh water, which is a true statement. Now consider the statement “No birds can swim.” This statement is false because at least one bird, the penguin, can swim. Therefore, the negation of this statement must be true. We may be tempted to write the negation as “All birds can swim,” but because this statement is also false it cannot be the negation. The correct negation m The Empire State Building in New York Holbox/Shutterstock
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