HW2CS40 - a Boolean circuit, using only AND, OR and NOT...

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Homework “Logic”, CS40, Summer 2008 15 points total due Tuesday, August 19, 11:00 am Question 1 (Truth tables, 3 points) . Write down in a systematic fashion the truth table of the following statement ϕ ( x , y , z ) = ( x y ) ( ¬ y ∧¬ z ) . Question 2 (“Laws of Logic”, 3 points) . Derive, using the 10 Laws of Logic (pp. 58– 59, Section 2.2 in the Reader), the following equivalence: ( p q ) ∧¬ ( q r ) ¬ q ∧¬ ( r ∨¬ p ) . Describe the steps that you use. Question 3 (Inference, 3 points) . Using the “Rules of Inference” (p. 76, Section 2.3 in the Reader), prove the implication (( p q ) t ) (( ¬ t r ) s ) (( p q ) ( r s )) . Describe the steps that you use. Question 4 (Designing circuits, 3 points) . Define the Boolean parity formula by P ARITY ( x 1 , ... , x 64 ) = ± 0 if x 1 + ··· + x 64 is even, 1 if x 1 + ··· + x 64 is odd. Describe
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Unformatted text preview: a Boolean circuit, using only AND, OR and NOT gates, that implements the function P ARITY as efficient as possible. (Efficient here means: with as few a number of gates as possible.) You do not have to draw the full circuit; it is sufficient to describe how you would draw it if you would have to. Question 5 (Formulas with quantifiers, 3 points) . Let f be a continuous function from the real numbers R to the real numbers R and consider the statement ∀ x ∃ y [ f ( x ) > y ] implies ∃ y ∀ x [ f ( x ) > y ] . Give an example of a function f that disproves this alleged implication. Explain briefly your line of reasoning behind your construction....
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This note was uploaded on 12/27/2011 for the course CMPSC 40 taught by Professor Egiceoclu during the Fall '09 term at UCSB.

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