C__work_280sol10

C__work_280sol10 - Solutions to Homework 10 May 1 2007 7.2...

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Unformatted text preview: Solutions to Homework 10 May 1, 2007 7.2: 3 The truth table for exclusive-or ( ⊕ ) and not ⇔ is shown below. P Q P ⊕ Q P ⇔ Q ¬ ( P ⇔ Q ) T T F T F T F T F T F T T F T F F F T F Nearly everyone received full credit here. 7.2: 10 This problem was perhaps a little unfair to those for whom English is not their native language, because it depends on the precise meanings of expressions with subtle differences in wording. But this is an important skill to learn—mathematical results are as often expressed in English as in mathematical logic. (e) This is slightly ambiguous. Normally, when people say this they mean “neither John nor Mary is here”, in which case it can be expressed by the wff ¬ J ∧ ¬ M (or any equivalent formula, ¬ ( J ∨ M ) for example). Sometimes people may interpret it like (f), that they are not both here (but maybe one of them is, or neither). This can be expressed as ¬ ( J ∧ M ). (f) There is only one correct interpretation of this, that no more than one of them is here, which we can write as ¬ ( J ∧ M ). A number of people interpreted it as meaning “exactly one of John and Mary is here”, which is not accepted usage. 1 (h) This is once again ambiguous—it could mean exclusive or (John is here or Mary is here, but not both), expressed as ¬ ( J ⇐⇒ M ) or ( J ∧ ¬ M ) ∨ ( ¬ J ∧ M ). To a mathematician, it would more naturally mean inclusive or (one or both of them are here), expressed as J ∨ M . (i) J ∧ ¬ M . (k) ¬ J ∨ ¬ M , or, equivalently ¬ ( J ∧ M ). 7.2, 12 (a) ( x = 2) ⇒ ( x 2 = 4), TRUE statement. (e) ( x > 0) ⇒ ( x 2 = 4 ⇒ x = 2), TRUE statement. (k) ( x 2 = 4 ⇒ x = 2) ∨ ( x 2 = 4 ⇒ x =- 2), TRUE statement. (l) (4 > 6) ⇔ (0 > 2), TRUE statement. Comments: The one and only of these that caused people prob- lems was (l). Heres a (plausible?) explanation for why just in case should be read as ⇔ : Let’s say that you didn’t know whether 4 > 6 is true or not. You could rationalize as follows: in case 4 > 6, then 0 > 2 by subtracting 4 from each side; otherwise, if the statement is false, then 0 > 2 must also be false (otherwise, it would be true that 4 > 6). Why is this useful? Well, there are some (many) unsolved conjectures in mathematics where you can say that if they are true, then something else results, otherwise it does not. For instance, let P be the statement that there is a polynomial time algorithm for deciding if a propositional formula is satisfiable and let Q be the statement that there is a polynomial time algorithm for deciding if a graph has a Hamiltonian path....
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C__work_280sol10 - Solutions to Homework 10 May 1 2007 7.2...

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