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math55-spring_2004-mt1-Demmel-soln

math55-spring_2004-mt1-Demmel-soln - Math 55 First Midterm...

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Unformatted text preview: Math 55 First Midterm 17 Feb 2004 NAME (1 pt): TA (1 pt): Name of Neighbor to your left (1 pt): Name of Neighbor to your right (1 pt): Instructions : This is a closed book, closed notes, closed calculator, closed computer, closed network, open brain exam. You get one point each for filling in the 4 lines at the top of this page. All other questions are worth 12 points. Write all your answers on this exam. If you need scratch paper, ask for it, write your name on each sheet, and attach it when you turn it in (we have a stapler). 1 2 3 4 5 Total 1 Question 1) (12 pts) Determine whether or not the following proposition is a tautology. a ⊕ b means the exclusive or of a and b . ( q ∧ ( p ⊕ r )) → ( p ∨ r ) If it is, prove it using rules for simplifying logical expressions, not using a truth table. If not, give a counterexample. Indicate what you are doing at each step (i.e. you don’t need to know the Latin names of inference rules, just convince us that you know what you’re doing). You may use the rule that ¬ ( a ⊕ b ) is logically equivalent to either ( ¬ a ) ⊕ b or a ⊕ ( ¬ b ). Answer: ( q ∧ ( p ⊕ r )) → ( p ∨ r ) ⇔ (definition of → ) ¬ ( q ∧ ( p ⊕ r )) ∨ ( p ∨ r ) ⇔ (DeMorgan’s Law) ( ¬ q ∨ ¬ ( p ⊕ r )) ∨ ( p ∨ r ) ⇔ (rule that ¬ ( p ⊕ r ) same as ( ¬ p ) ⊕ r ) ( ¬ q ∨ ( ¬ p ⊕ r )) ∨ ( p ∨ r ) ⇔ (definition of ⊕ ) ( ¬ q ∨ (( ¬ p ∧ ¬ r ) ∨ ( ¬¬ p ∧ r )) ∨ ( p ∨ r ) ⇔ (double negative) ( ¬ q ∨ (( ¬ p ∧ ¬ r ) ∨ ( p ∧ r )) ∨ ( p ∨ r ) ⇔ (DeMorgan’s Law) ( ¬ q ∨ ( ¬ ( p ∨ r ) ∨ ( p ∧ r )) ∨ ( p ∨ r ) ⇔ (associativity and commutativity of ∨ ) (( p ∨ r ) ∨ ( ¬ ( p ∨ r )) ∨ ( p ∧ r ) ∨ ¬ q ⇔ ( x ∨ ¬ x is always true, where x = p ∨ r ) ( T ) ∨ ( p ∧ r ) ∨ ¬ q ⇔ ( T ∨ x is always true, for any x ) T 2 Question 1) (12 pts) Determine whether or not the following proposition is a tautology. c ⊕ d means the exclusive or of c and d . (( t ⊕ r ) ∧ ¬ s ) → ( t ∨ r ) If it is, prove it using rules for simplifying logical expressions, not using a truth table. If not, give a counterexample. Indicate what you are doing at each step (i.e. you don’t need to know the Latin names of inference rules, just convince us that you know what you’re doing). You may use the rule that ¬ ( c ⊕ d ) is logically equivalent to either ( ¬ c ) ⊕ d or c ⊕ ( ¬ d ). Answer: ( ¬ s ∧ ( t ⊕ r )) → ( t ∨ r ) ⇔ (definition of → ) ¬ ( ¬ s ∧ ( t ⊕ r )) ∨ ( t ∨ r ) ⇔ (DeMorgan’s Law) ( ¬¬ s ∨ ¬ ( t ⊕ r )) ∨ ( t ∨ r ) ⇔ (double negative) ( s ∨ ¬ ( t ⊕ r )) ∨ ( t ∨ r ) ⇔ (rule that ¬ ( t ⊕ r ) same as ( ¬ t ) ⊕ r ) ( s ∨ ( ¬ t ⊕ r )) ∨ ( t ∨ r ) ⇔ (definition of ⊕ ) ( s ∨ (( ¬ t ∧ ¬ r ) ∨ ( ¬¬ t ∧ r )) ∨ ( t ∨ r ) ⇔ (double negative) ( s ∨ (( ¬ t ∧ ¬ r ) ∨ ( t ∧ r )) ∨ ( t ∨ r ) ⇔ (DeMorgan’s Law) ( s ∨ ( ¬...
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math55-spring_2004-mt1-Demmel-soln - Math 55 First Midterm...

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