74ShwS - Math 74 – Homework Solutions Adam Booth Spring...

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Unformatted text preview: Math 74 – Homework Solutions. Adam Booth. Spring 2007. 1.1 3 Let A be the statement “Alice is in the room” and B be the statement “Bob is in the room.” (a) ¬ ( A ∧ B ). (b) ¬ A ∧ ¬ B . (c) ¬ A ∨ ¬ B . (d) ¬ ( a ∨ B ). 4 (a) is; (b) isn’t as the comma is a not a symbol in our language; (c) is; (d) isn’t as there’s not connective between the two parenthesesed expressions. 6 (a) One out of Steve and George is happy and one is unhappy. (b) Either George is unhappy or otherwise it’s either the case that Steve is happy or that both George is happy and Steve is unhappy. (c) Either Steve is happy or it’s the case both that George is happy and that one of Steve and George is unhappy. 1.2 2 (a) P Q ¬ P Q ∨ ¬ P P ∧ ( Q ∨ ¬ P ) ¬ ( P ∧ ( Q ∨ ¬ P )) F F T T F T F T T T F T T F F F F T T T F T T T (b) P Q R P ∨ Q ¬ P ¬ P ∨ R ( P ∨ Q ) ∧ ( ¬ P ∨ R ) F F F F T T F F F T F T T F F T F T T T T F T T T T T T T F F T F F F T F T T F T T T T F T F F F T T T T F T T 1 3 (a) P Q P + Q F F F F T T T F T T T F (b) One such formula is ( P ∧¬ Q ) ∨ ( ¬ P ∧ Q ). The truth table turns out the same. 6 (a) P Q P | Q F F T F T T T F T T T F (b) One such formula is ¬ ( P ∧ Q ). (c) We have ¬ P ≡ ( P | P ). As we know from (b) that P ∧ Q ≡ ¬ ( P | Q ) we can apply this to get P ∧ Q ≡ (( P | Q ) | ( P | Q )). We also know that P ∨ Q ≡ ¬ ( ¬ P ∧¬ Q ), so putting this together with the last two results we get that P ∨ Q ≡ (((( P | P ) | ( Q | Q )) | (( P | P ) | ( Q | Q ))) | ((( P | P ) | ( Q | Q )) | (( P | P ) | ( Q | Q )))) . You can see why using | as our only connective would be possible but rather impractical! 8 Let’s examine the truth tables. P Q ( P ∧ Q ) ∨ ( ¬ P ∧ ¬ Q ) ¬ P ∧ Q ( P ∧ ¬ Q ) ∨ ( ¬ P ∧ Q ) ¬ ( P ∨ Q ) ( Q ∧ P ) ∨ ¬ P F F T T T T T F T F T T F T T F F F T F F T T T T T F T We observe that (b) and (e) are equivalent; none of the rest equivalent to any of the others. 10 (a) P Q ¬ ( P ∨ Q ) ¬ P ∧ ¬ Q F F T T F T F F T F F F T T F F (b) P Q R P ∧ ( Q ∨ R ) ( P ∧ Q ) ∨ ( P ∧ R ) F F F F F F F T F F F T F F F F T T F F T F F F F T F T T T T T F T T T T T T T The other distribution law is similar. 2 12 (a) ¬ ( ¬ P ∨ Q ) ∨ ( P ∧ ¬ R ) ≡ ( P ∧ ¬ Q ) ∨ ( P ∧ ¬ R ) DeMorgan and Double Negation ≡ P ∧ ( ¬ Q ∨ ¬ R ) Distribution. (b) ¬ ( ¬ P ∧ Q ) ∨ ( P ∧ ¬ R ) ≡ ( P ∨ ¬ Q ) ∨ ( P ∧ ¬ R ) DeMorgan and Double Negation ≡ ¬ Q ∨ ( P ∨ ( P ∧ ¬ R )) Commutativity and Associativity ≡ ¬ Q ∨ P Absorption. (c) ( P ∧ R ) ∨ [ ¬ R ∧ ( P ∨ Q )] ≡ ( P ∧ R ) ∨ [( P ∧ ¬ R ) ∨ ( ¬ R ∧ Q )] Distribution ≡ [ P ∧ ( R ∨ ¬ R )] ∨ ( P ∧ Q ) Associativity, Distribution ≡ P ∨ ( ¬ R ∧ Q ) Tautology law....
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This note was uploaded on 04/17/2008 for the course MATH 74 taught by Professor Courtney during the Spring '07 term at Berkeley.

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74ShwS - Math 74 – Homework Solutions Adam Booth Spring...

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