3.4 Equivalent Statements 135 Consider p q p q ( ) , ∼ ∧ ⇔∼ ∨ ∼ one of De Morgan’s laws. To go from p q ( ) ∼ ∧ to p q, ∼ ∨ ∼ we negate both the p and the q within parentheses; change the conjunction, ,∧ to a disjunction, ;∨ and remove the negation symbol preceding the left parentheses and the parentheses themselves. We can use a similar procedure to obtain equivalent statements. For example, p q p q ( ) ∼ ∼ ∧ ⇔ ∨ ∼ p q p q ( ) ∼ ∧ ∼ ⇔∼ ∨ We can use a similar procedure to obtain equivalent statements when a disjunction is within parentheses. Note that p q p q ( ) ∼ ∼ ∨ ⇔ ∧ ∼ p q p q ( ) ∼ ∨ ∼ ⇔∼ ∧ Example 5 Use De Morgan’s Laws Write a statement that is logically equivalent to “It is false that you can take Trigonometry to meet your mathematics requirement or you can take Biology to meet your science requirement.” Solution Let p q : You can take Trigonometry to meet your mathematics requirement. : You can take Biology to meet your science requirement. The given statement is of the form p q ( ). ∼ ∨ Using the second of De Morgan’s laws, we see that an equivalent statement in symbols is p q. ∼ ∧ ∼ Therefore, an equivalent statement is “You cannot take Trigonometry to meet your mathematics requirement and you cannot take Biology to meet your science requirement.” 7 Now try Exercise 25 Example 6 Using De Morgan’s Laws Use De Morgan’s Laws to write a statement logically equivalent to “We cannot go to the parade, but we can go to the amusement park.” Solution Let p q : We can go to the parade. : We can go to the amusement park. The statement written symbolically is p q. ∼ ∧ Earlier we showed that p q p q ( ) ∼ ∧ ⇔∼ ∨ ∼ Therefore, the statement “It is false that we can go to the parade or we cannot go to the amusement park” is logically equivalent to the given statement. 7 Now try Exercise 27 There are strong similarities between the topics of set theory and logic. We can see them by examining De Morgan’s laws for sets and logic. De Morgan’s laws: set theory A B A B ( ) ∩ ′ = ′ ∪ ′ A B A B ( ) ∪ ′ = ′ ∩ ′ De Morgan’s laws: logic p q p q ( ) ∼ ∧ ⇔∼ ∨ ∼ p q p q ( ) ∼ ∨ ⇔∼ ∧ ∼ The complement in set theory, ,′ is similar to the negation, ,∼ in logic. The intersection, ,∩ is similar to the conjunction, ;∧ and the union, ,∪ is similar to the disjunction, .∨ Degree/eStock Photo/ Alamy Stock Photo Continued on next page
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