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Homework 01

# Homework 01 - (j If Steve eats a doughnut then it rocks him...

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MATH 74 HOMEWORK 1 DAN BERWICK EVANS The following is due Wednesday, January 28th at 3:10 pm. (1) (Eccles 1.2) Construct truth tables for the statements: (a) not ( P and Q ) (b) (not P ) or (not Q ) (c) P and (not Q ) (d) (not P ) or Q . (2) Convert the following sentences into the notation of propositional and predicate logic, using the given list of predicates. The first problem is done as an example. Predicates: T ( x ) = “ x is tired.” B ( x ) = x goes to bed early.” C ( x ) =“ x is cranky.” L ( x, y ) = “ x likes y .” W ( x, y ) = “ x wants y.” E ( x, y ) = x eats y .” R ( x, y ) = “ x rocks y like a hurricane.” (a) If Steve is tired, then he goes to bed early. Solution: T (Steve) = B (Steve). (b) Steve is tired and cranky. (c) Steve only goes to bed early if he is tired. (d) Steve likes doughnuts and coffee. (e) Steve likes either doughnuts or coffee (possibly both!). (f) Steve likes either doughnuts or coffee, but not both. (g) Steve only likes coffee if he is cranky. (h) Steve eats doughnuts whenever he wants doughnuts. (i) If Steve eats doughnuts and plums, then Steve is tired and cranky and goes to bed early.
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Unformatted text preview: (j) If Steve eats a doughnut, then it rocks him like a hurricane. (3) (Eccles 1.3) Using the truth table for ‘or’ complete the following truth table for the statement a ≤ b : a b a < b a = b a ≤ b 1 1 2 1 1 2-2 1 (4) (Eccles 1.5) Prove that | a | 2 = a 2 for every real number a . (5) (Bergman Exercise 3) Suppose P and Q are two mathematical assertions. (Examples might be “ n > 0” and “ n is even” if we are talking about an integer n .) For each statement in the left-hand column, ﬁnd the logically equivalent statement in the right-hand column. (There are two statements in the left-hand column which are not equivalent to any statements in the right-hand column.) P ∧ Q Q = ⇒ P P ∨ ( ¬ Q ) Q ∧ P P ( ¬ P ) = ⇒ Q P ∨ Q ( ¬ P ) ∧ ( ¬ Q ) ¬ ( P ∧ Q ) ( ¬ P ) ∨ ( ¬ Q ) ¬ P ¬¬ P 1...
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