136 CHAPTER 3 Logic Conditional Statements as Disjunctions Statements containing connectives other than and and or may have equivalent statements. To illustrate this point, construct truth tables for p q → and for p q. ∼ ∨ The truth tables will have the same answer columns and therefore the statements are equivalent. We summarize this as follows. If we were to interchange the set symbols with the logic symbols, De Morgan’s laws would remain, but in a different form. Both ′ and ∼ can be interpreted as not . Both ∩ and ∧ can be interpreted as and . Both ∪ and ∨ can be interpreted as or . For example, the set statement A B ′ ∪ can be written as a statement in logic as a b. ∼ ∨ The Conditional Statement Written as a Disjunction p q p q → ⇔∼ ∨ With these equivalent statements, we can write a conditional statement as a disjunction or a disjunction as a conditional statement. For example, the statement “If the game is polo, then you ride a horse” can be equivalently stated as “The game is not polo or you ride a horse.” To change a conditional statement to a disjunction, negate the antecedent, change the conditional symbol to a disjunction symbol, and keep the consequent the same. To change a disjunction statement to a conditional statement, negate the first statement, change the disjunction symbol to a conditional symbol, and keep the second statement the same. Example 7 Rewriting a Disjunction as a Conditional Statement Write a conditional statement that is logically equivalent to “The cows are in the pasture or the horses are not in the barn.” Solution Let p q : The cows are in the pasture. : The horses are in the barn. The original statement may be written symbolically as p q. ∨ ∼ To write an equivalent conditional statement, negate the first statement, p; replace the disjunction symbol, ,∨ with a conditional symbol, ;→ and keep the second statement the same. Symbolically, the equivalent statement is p q. ∼ →∼ The equivalent statement in words is “If the cows are not in the pasture, then the horses are not in the barn.” 7 Now try Exercise 33 Negation of the Conditional Statement Now we will discuss how to negate a conditional statement. To negate a conditional statement we use the fact that p q p q → ⇔∼ ∨ and De Morgan’s laws. Examples 8 and 9 show this process. Profile in Mathematics Charles Dodgson One of the more interesting and well-known students of logic was Charles Dodgson (1832–1898), better known to us as Lewis Carroll, the author of Alice’s Adventures in Wonderland and Through the Looking-Glass . Although the books have a child’s point of view, many argue that the audience best equipped to enjoy them is an adult one. Dodgson, a mathematician, logician, and photographer (among other things), uses the naïveté of a 7-year-old girl to show what can happen when the rules of logic are taken to absurd extremes. “You should say what you mean,” the March Hare went on. “I do,” Alice hastily replied; “at least—at least I mean what I say— that’s the same thing, you know.” “Not the same thing a bit!” said the Hatter. “You might as well say that ’I see what I eat’ is the same thing as ’I eat what I see’!” Doodeez/Shutterstock
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