Solutions04

# Solutions04 - order to make the picture ﬁt on the page...

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Chapter 4: Hints and Selected Solutions Section 4.1 (page 104) 4.2 1. The truth table for ( A B ) ( ¬ A ∨ ¬ B ) is shown in Figure ?? . Since all the entries under the main connective ( ) are T, it shows that the sentence is a tautology. 4.5 The truth table is shown below. Since not all the entries under the main connective are T, the sentence is not a tautology. However, some of the entries in this column are T, so the sentence is TT-possible. 1

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4.9 Make sure you understand why there could not be a ”Yes” in the first column and a ”No” in the second. If you do not understand this, be sure to look it up in the book or ask your instructor. It is a simple but key point. Here are a few of the answers for this problem, so you can see if you have the right idea. Sentence tw -possible tt -possible 1 Yes Yes 4 No Yes 7 No Yes 10 Yes Yes Section 4.2 (page 109) 4.12 The truth table is shown below. Since the truth values under the main connectives of each sentence are the same, row by row, it shows that the sentences are tautologically equivalent. 2
Section 4.3 (page 113) 4.21 The truth table is shown below. (We have abbreviated the sentences in

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Unformatted text preview: order to make the picture ﬁt on the page. You should not abbreviate yours, or GG will complain.) Since there is a row (the ﬁrst) where the ﬁrst two sentences are both true and the third is false, it shows that the third is not a tautological consequence of the ﬁrst two. Section 4.4 (page 116) 3 4.26 The complete proofs of Taut Con 1 and Taut Con 2 are shown below. Section 4.5 (page 120) 4.31 2. ¬ Cube ( a ) ∨ ¬ Larger ( b , a ) 4. ¬ Cube ( a ) ∧ Larger ( b , a ) 6. Cube ( a ) ∧ Larger ( b , a ) ∧ a = b 8. ¬ Tet ( b ) ∧ ( ¬ Large ( c ) ∨ Smaller ( d , e )) 10. Dodec ( f ) ∨ ( ¬ Tet ( b ) ∧ Tet ( f ) ∧ Dodec ( f )) (This sentence is logically equivalent to Dodec ( f ), so that would be a good answer, too.) 4 4.33 ( A ∧ B ) ∧ A ⇔ A ∧ ( B ∧ A ) Associativity of ∧ ⇔ A ∧ ( A ∧ B ) Commutivity of ∧ ⇔ ( A ∧ A ) ∧ B Associativity of ∧ ⇔ A ∧ B Idempotence of ∧ 5...
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