ICS253_081_Major1

ICS253_081_Major1 - p and q are false, and it is false...

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ICS 253: Discrete Structures 1 (Sections 1 and 2) Adel F. Ahmed 6 November 2008 (081) Major Exam 1 Duration 120 minutes Question Max Pts Points 1 20 2 40 3 40 Total 100 St.Name: St.Number:
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Question 1: [20 points] a) [5 points] Translate the nested quantification into an English statement that expresses a math- ematical fact. The domain in this case consists of all real numbers. x y ((( x < 0) ( y < 0)) ( xy > 0)) b) [15 points] Determine whether ( ¬ q ( p q )) → ¬ p is a tautology without using the truth tables. [Indicate which rule was used at each step in your solution.]
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Question 2: [40 points] a) [10 points] Express the negation of the following statement so that all negation symbols imme- diately precede predicates. x yP ( x,y ) ∨∀ x yQ ( x,y ) b) [10 points] Find a counterexample, if possible, to the following statements, where the domain for all variables consists of all integers. 1. x ( x 2 x ) 2. x y ( y 2 = x ) 3. x ( x = 1) 4. x y ( x 2 = y 2 x = y ) c) [20 points] Let us define a new logical operator denoted by the symbol ± , where the proposition p ± q is true when both
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Unformatted text preview: p and q are false, and it is false otherwise. 1. Show that p q is logically equivalent to ( p q ). 2. If p p p and ( p q ) ( p q ) p q then write the compound proposition logically equivalent to p q using only the logical operator . Question 3: [40 points] a) [15 points] Let M ( x,y ) x has sent y and e-mail message, where the domain consists of all students in your class. Use quantiers and logical connectives to express the following statement: There are two dierent students in your class who have sent each other e-mail messages. b) [25 points] Use rules of inference to show that if x ( P ( x ) Q ( x )) and x ( ( P ( x ) Q ( x )) R ( x ) ) are true, then x ( R ( x ) P ( x )) is also true, where the domain of all quantiers are the same. [Indicate which rule was used at each step in your solution.] Scribble paper:...
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This note was uploaded on 01/27/2011 for the course ICS 253 taught by Professor Arnasser during the Spring '10 term at GWU.

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ICS253_081_Major1 - p and q are false, and it is false...

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