07au366finalprsolutions

07au366finalprsolutions - Solutions to Selected Math 366...

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Unformatted text preview: Solutions to Selected Math 366 Final Exam Practice Problems Autumn 2005 1. Each of the following statements has one of the forms ∼ p p ∧ q p ∨ q p → q p ↔ q Find the appropriate form and indicate what each statement variable in your choice represents. (a) If Archibald passes the first exam, then he will not drop the course. Solution. The statement has form p → q where p = ”Archibald passes the first exam.” q = ”Archibald will not drop the course.” 2. Use truth tables to verify each of the following logical equivalences. (a) p ∨ ( ∼ p ∧ q ) ≡ p ∨ q Solution. p q ∼ p ∼ p ∧ q p ∨ ( ∼ p ∧ q ) p ∨ q T T F F T T T F F F T T F T T T T T F F T F F F Since p ∨ ( ∼ p ∧ q ) and p ∨ q have the same truth table, p ∨ ( ∼ p ∧ q ) ≡ p ∨ q . 3. Show that each of the following arguments has a valid argument form by exhibiting such a form. Explain what each statement variable in your form represents. (b) If Christine intends to go to the party, then John will also. John is not intending to go to the party. Therefore, Christine is not intending to go to the party. Solution. The argument has the form p → q ∼ q . . . ∼ q where p = ”Christine intends to go to the party.” q = ”John will go to the party.” This form is valid since it is an instance of Modus Tolens. 4. Determine which of the following argument forms are valid and which are not. Justify your answers. If the form is valid, verify that it is by two methods: truth tables and step by step derivations using theorem 1.1.1 and table 1.3.1 from the text. 1 (b) p → ( q → r ) ∼ r p . . . ∼ q Solution. Premises: p → ( q → r ) ∼ r p 1. p → ( q → r ) premise p premise . . . q → r by MP 2. q → r from 1 ∼ r premise . . . ∼ q by MT 5. Find a Boolean expression which has the following I/O table. P Q R ouput 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 The recognizers for the rows with output 1 are P ∧ Q ∧ R , P ∧ ∼ Q ∧ ∼ R and ∼ P ∧ ∼ Q ∧ R . Therefore, ( P ∧ Q ∧ R ) ∨ ( P ∧ ∼ Q ∧ ∼ R ) ∨ ( ∼ P ∧ ∼ Q ∧ R ) has the above I/O table....
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This note was uploaded on 07/17/2008 for the course MATH 366 taught by Professor Joshua during the Winter '08 term at Ohio State.

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07au366finalprsolutions - Solutions to Selected Math 366...

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