126 CHAPTER 3 Logic b) Motorola has 18% of the U.S. market share or others have 20% of the U.S. market share, if and only if Samsung has 29% of the U.S. market share. Solution a) Let p: Apple has 57% of the U.S. market share. q: Samsung has 36% of the U.S. market share. r: LG has 2.5% of the U.S. market share. Then the original compound statement can be written with symbols as p q r ( ) . ∧ → From Fig. 3.2 we can determine that both p and r are true statements and that q is a false statement. We substitute T for p, F for q, and T for r and evaluate the compound statement. p q r F T T ( ) (T ) F T ∧ → ∧ → → Therefore, the original compound statement “If Apple has 57% of the U.S. market share and Samsung has 36% of the U.S. market share, then LG has 2.5% of the U.S. market share” is true. b) Let p: Motorola has 18% of the U.S. market share. q: Others have 20% of the U.S. market share. r: Samsung has 29% of the U.S. market share. Then the original compound statement can be written with symbols as p q r ( ) . ∨ ↔ From Fig. 3.2 we can determine that both p and q are false statements and that r is a true statement. We substitute F for p, F for q, and T for r and evaluate the compound statement. p q r ( ) (F F) T F T F ∨ ↔ ∨ ↔ ↔ Therefore, the original compound statement “Motorola has 18% of the U.S. market share or others have 20% of the U.S. market share, if and only if Samsung has 29% of the U.S. market share” is a false statement. 7 Now try Exercise 61 RECREATIONAL MATH Satisfiability Problems Suppose you are hosting a dinner party for seven people: Yasumasa, Marie, Albert, Stephen, Leonhard, Karl, and Emmy. You need to develop a seating plan around your circular dining room table that would satisfy all your guests. Albert and Emmy are great friends and must sit together. Yasumasa and Karl haven’t spoken to each other in years and cannot sit by each other. Leonhard must sit by Marie or by Albert, but he cannot sit by Karl. Stephen insists on sitting by Albert. Can you come up with a plan that would satisfy all your guests? Now imagine the difficulty of such a problem as the list of guests, and their demands, grows. Problems such as this are known as satisfiability problems. The symbolic logic you are studying in this chapter allows computer scientists to represent these problems with symbols and solve the problems using computers. Even with the fastest computers, some satisfiability problems take an enormous amount of time to solve. One solution to the problem posed above is shown upside down below. Exercises 74 and 75 have other satisfiability problems. Albert Emmy Yasumasa Leonhard Marie Karl Stephen Self-Contradictions, Tautologies, and Implications Two special situations can occur in the answer column of a truth table of a compound statement: The statement may always be false, or the statement may always be true. We give such statements special names. Definition: Self-contradiction A self-contradiction is a compound statement that is always false. When every truth value in the answer column of the truth table is false, then the statement is a self-contradiction.
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