100 CHAPTER 3 Logic Compound Statements Statements consisting of two or more simple statements are called compound statements . The connectives often used to join two simple statements are … … and or if then if and only if , , , In addition, we consider a simple statement that has been negated to be a compound statement. The word not is generally used to negate a statement. To reduce the amount of writing in logic, it is common to represent each simple statement with a lowercase letter. For example, suppose we are discussing the simple statement “Leland is a farmer.” Instead of writing “Leland is a farmer” over and over again, we can let p represent the statement “Leland is a farmer.” Thereafter we can simply refer to the statement with the letter p. It is customary to use the letters p q r , , , and s to represent simple statements, but other letters may be used instead. Let’s now look at the connectives used to make compound statements. Not Statements The negation is symbolized by ∼ and read “not.” For example, the negation of the statement “Steve is a college student” is “Steve is not a college student.” If p represents the simple statement “Steve is a college student,” then p∼ represents the compound statement “Steve is not a college student.” For any statement p p p , ( ) . ∼ ∼ = For example, the negation of the statement “Steve is not a college student” is “Steve is a college student.” Consider the statement “Inga is not at home.” This statement contains the word not, which indicates that it is a negation. To write this statement symbolically, we let p represent “Inga is at home.” Then p∼ would be “Inga is not at home.” We will use this convention of letting letters such as p, q, or r represent statements that are not negated. We will indicate that a statement is negated with the negation symbol, .∼ And Statements The conjunction is symbolized by ∧ and read “and.” The ∧ looks like an A (for And) with the bar missing. Let p and q represent the simple statements. p q : You will perform 5 months of community service. : You will pay a $100 fine. Then the following is the conjunction written in symbolic form. p q You will perform 5 months of community service and you will pay a $100 fine. . ∧ The conjunction is generally expressed as and . Other words sometimes used to express a conjunction are but, however, and nevertheless . Example 2 Write a Conjunction The following statement involves Jon Batiste, an American jazz musician, band leader, and television personality. Write the following conjunction in symbolic form. Jon Batiste is the leader of the band Stay Human, but Jon Batiste is not playing the Chicago Jazz Festival. Solution Let l and p represent the following simple statements. l: Jon Batiste is the leader of the band Stay Human. p: Jon Batiste is playing the Chicago Jazz Festival. In symbolic form, the compound statement is l p. ∧ ∼ 7 Now try Exercise 27 RECREATIONAL MATH Sudoku Solving puzzles requires us to use logic. Sudoku is a puzzle that originated in Japan and continues to gain popularity worldwide. To solve the puzzle, you need to place every digit from 1 to 9 exactly one time in each row, in each column, and in each of the nine 3 by 3 boxes. For more information and a daily puzzle see www.web sudoku.com. Furthermore, there are many apps that can be used to play Sudoku on your tablet or smartphone. The solution to the puzzle above can be found in the Answers section in the back of this book. For an additional puzzle see Exercise 82 on page 107. 1 4 8 5 6 5 3 8 6 9 1 5 7 2 4 2 3 6 4 5 1 4 9 2 5 2 7 8 3 9 1 1 4 7 9 8
RkJQdWJsaXNoZXIy NjM5ODQ=