134 CHAPTER 3 Logic Now try Exercise 15 Table 3.29 (a) (b) (c) (d) p q q∼ → p∼ p∼ ∨ q∼ q∼ ∧ p∼ p∼ → q∼ TTFTFFFFFFFFTF TFTFFFTTTFFFTT FTFTTTTFFFTTFF FFTTTTTTTTTTTT 7 De Morgan’s Laws for Logic 1. p q p q ( ) ∼ ∧ ⇔∼ ∨ ∼ 2. p q p q ( ) ∼ ∨ ⇔∼ ∧ ∼ De Morgan’s Laws for Logic Example 3 showed that a statement of the form p q ( ) ∼ ∧ is equivalent to a statement of the form p q. ∼ ∨ ∼ Thus, we may write p q p q ( ) . ∼ ∧ ⇔∼ ∨ ∼ This equivalent statement is the first of two special laws called De Morgan’s laws. The laws, named after Augustus De Morgan, an English mathematician, were first introduced in Section 2.4, where they applied to sets. You can demonstrate that De Morgan’s second law is true by constructing and comparing truth tables for p q ( ) ∼ ∨ and p q. ∼ ∧ ∼ Profile in Mathematics Augustus De Morgan Augustus De Morgan (1806– 1871), the son of a member of the East India Company, was born in India and educated at Trinity College, Cambridge (UK). One of the great reformers of logic in the nineteenth century, De Morgan made his greatest contribution to the subject by realizing that logic as it had come down from Aristotle was narrow in scope and could be applied to a wider range of arguments. His work laid the foundation for modern, symbolic logic. When using De Morgan’s laws, if it becomes necessary to negate an already negated statement, use the fact that p ( ) ∼ ∼ is equivalent to p. For example, the negation of the statement “Today is not Monday” is “Today is Monday.” Example 4 Use De Morgan’s Laws Select the statement that is logically equivalent to “I do not have investments, but I do not have debts.” a) I do not have investments or I do not have debts. b) It is false that I have investments and I have debts. c) It is false that I have investments or I have debts. d) I have investments or I have debts. Solution To determine which statement is equivalent, write each statement in symbolic form. Let p q : I have investments. : I have debts. The statement “I do not have investments, but I do not have debts” written symbolically is p q. ∼ ∧ ∼ Recall that the word but means the same thing as and . Now, write parts (a) through (d) symbolically. a) p q ∼ ∨ ∼ b) p q ( ) ∼ ∧ c) p q ( ) ∼ ∨ d) p q ∨ De Morgan’s law shows that p q ∼ ∧ ∼ is equivalent to p q ( ). ∼ ∨ Therefore, the answer is (c): “It is false that I have investments or I have debts.” 7 Now try Exercise 17
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