102 CHAPTER 3 Logic The information provided in Example 4 is summarized below. Statement Symbolic representation Type of statement Dinner includes soup, and salad or the vegetable of the day. ∧ ∨ p q r ( ) conjunction Dinner includes soup and salad, or the vegetable of the day. ∧ ∨ p q r ( ) disjunction A negation symbol has the effect of negating only the statement that directly follows it. To negate a compound statement, we must use parentheses. When a negation symbol is placed in front of a statement in parentheses, it negates the entire statement in parentheses. The negation symbol in this case is read, “It is not true that …” or “It is false that …” Solution a) The comma tells us to group the statement “Dinner includes salad” with the statement “Dinner includes the vegetable of the day.” Note that both statements are on the same side of the comma. The statement in symbolic form is ∧ ∨ p q r ( ). In mathematics, we always evaluate the information within the parentheses first. Since the conjunction, ∧, is outside the parentheses and is evaluated last, this statement is considered a conjunction. b) The comma tells us to group the statement “Dinner includes soup” with the statement “Dinner includes salad.” Note that both statements are on the same side of the comma. The statement in symbolic form is ∧ ∨ p q r ( ) . Since the disjunction, ∨, is outside the parentheses and is evaluated last, this statement is considered a disjunction. 7 Now try Exercise 49 Example 5 Change Symbolic Statements into Words Let p q : We will go to the beach. : We will go to the park. Write the following symbolic statements in words. a) p q ∧ ∼ b) p q ∼ ∨ ∼ c) p q ( ) ∼ ∧ Solution a) We will go to the beach and we will not go to the park. b) We will not go to the beach or we will not go to the park. c) It is false that we will go to the beach and we will go to the park. 7 Now try Exercise 41 Recall that the word but may also be used in a conjunction. Therefore, Example 5(a) could also be written “We will go to the beach, but we will not go to the park.” Part (b) of Example 5 is a disjunction, since it can be written p q ( ) ( ). ∼ ∨ ∼ Part (c), which is p q ( ), ∼ ∧ is a negation, since the negation symbol negates the entire statement within parentheses. The similarity of these two statements is discussed in Section 3.4. Occasionally, we come across a neither–nor statement, such as “John is neither an accountant nor a football fan.” This statement means that “John is not an accountant” and “John is not a football fan.” If p represents “John is an accountant” and q represents “John is a football fan,” this statement is symbolized by p q . ∼ ∧ ∼ Mark Bowden/123RF
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