h1-sol

h1-sol - x = y 2(False Suppose x = a satis±es the above...

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Solutions of HW#1 : Basic Logic Q.1 Make truth tables for the following statement: ( p q ) ( q p ) Answer p q p q q p ( p q ) ( q p ) T T T T T T F F T T F T T F F F F T T T ( p q ) ( ¬ p q ) Answer p q p q ¬ p q ( p q ) ( ¬ p q ) T T T T T T F F T F F T T T T F F T F F Q.2 Using logical equivalences discussed in class prove that ( p q ) ( p q ) is a tautology, that is, prove that ( p q ) ( p q ) T. Answer ( p q ) ( p q ) ≡ ¬ ( p q ) ( p q ) ( ¬ p ∨ ¬ q ) ( p q ) ( ¬ p p ) ( ¬ q q ) T T T Note: Another way to solve this question is by constructing the truth table for the given logical expression and showing that it always yields T for all values of p and q . 1

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Q.3 Let Determine the truth value of the following statements when x and y are real numbers: 1. x y ( x = y 2 ), 2. x y ( xy = 0), 3. x n =0 y xy = 1, 4. x y ( x + y n = y + x ). Answer 1. x y
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Unformatted text preview: ( x = y 2 ): (False) Suppose x = a satis±es the above statement. This means that a = y 2 for all real values of y . This is impossible since y would then be equal at most 2 real values ( ± √ a ). 2. ∃ x ∀ y ( xy = 0): (True) At x = 0, xy = 0 for all real values of y . 3. ∀ x n =0 ∃ y xy = 1: (True) If we pick an arbitrary value for x n = 0, say a , then there exists a value for y that satis±es the given statement, y = 1 a . 4. ∃ x ∃ y ( x + y n = y + x ): (False) The addition of real numbers is always commutative operation. 2...
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h1-sol - x = y 2(False Suppose x = a satis±es the above...

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