166 CHAPTER 3 Logic 25. p p p q r q r , ∨ ∼ ∧ ∧ ∧ ∧ p p q r p q r ( ) ; ; not equivalent 26. r p q r p r , p r q q ⎡⎣ ∧ ∨ ∧∼⎤⎦∨ ∧∼ ∨ ∧ ∨ p q r q r p p r q r ( ) ( ) ( ); ( ) ( ); equivalent Concept/Writing Exercises 27. Explain why the lightbulb will never go on in the following circuit. * p p 28. Explain why the lightbulb will always be on in the following circuit. * p p Challenge Problems/Group Activities 29. Design a circuit that can be represented by a) p q → * b) p q ( ) ∼ → * (Hint: See Section 3.4.) 30. Design two circuits that each involve switches labeled p q r , , , and s that appear to be different but are actually equivalent circuits. Answers will vary. Research Activities 31. Electronic Gates Digital computers use gates that work like switches to perform calculations. (See Did You Know? box on page 164.) Information is fed into the gates and information leaves the gates, according to the type of gate. The three basic gates used in computers are the NOT gate, the AND gate, and the OR gate. Do research on the three types of gates. a) Explain how each gate works. b) Explain the relationship between each gate and the corresponding logic connectives not, and, and or. c) Illustrate how two or more gates can be combined to form a more complex gate. *See Instructor Answer Appendix Summary Chapter 3 Important Facts and Concepts Examples Section 3.1 Quantifiers Form of Statement Form of Negation All are. Some are not. None are. Some are. Some are. None are. Some are not. All are. Summary of Connectives Formal Name Symbol Read Symbolic Form Negation ∼ not p~ Conjunction ∧ and ∧ p q Disjunction ∨ or ∨ p q Conditional → if–then →p q Biconditional ↔ if and only if ↔p q Example 1, page 99 Examples 2–8, pages 100–104
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