# hmw1 - b(01111 âˆ 10101 âˆ¨ 01000 = 00101 âˆ¨ 01000 = 01101...

This preview shows pages 1–2. Sign up to view the full content.

Solutions to Homework 1. Math 55, Fall 2006. Prob 1.1.24. (a) Converse: If I stay home, then it will snow tonight. Contrapositive: If I do not stay at home, then it will not snow tonight. (b) Converse: Whenever I go to the beach, it is a sunny summer day. Con- trapositive: Whenever I do not go to the beach, it is not a sunny summer day. (c) Converse: If I speel until noon, then I stayed up late. Contrapositive: If I do not sleep until noon, then I did not stay up late. Prob 1.1.38. For parts (a) and (b) we have the following table: p p p ¬ p p ⊕ ¬ p T F F T F F T T For parts (c) and (D) the table is below (columns ﬁve and six): p q ¬ p ¬ q p ⊕ ¬ q ¬ p ⊕ ¬ q F F T T T F F T T F F T T F F T F T T T F F T F For parts (e) and (f) we have the following table (columns ﬁve and six). This time the column showing the negation of q is omitted. Note that the statement (e) is a tautology and the statement (f) is a contradiction. p q p q p ⊕ ¬ q ( p q ) ( p ⊕ ¬ q ) ( p q ) ( p ⊕ ¬ q ) F F F T T F F T T F T F T F T F T F T T F T T F Prob 1.1.38. ( a ) 11000 (01011 11011) = 11000 11011 = 11000

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: ( b ) (01111 âˆ§ 10101) âˆ¨ 01000 = 00101 âˆ¨ 01000 = 01101 ( c ) (01010 âŠ• 11011) âŠ• 01000 = 10001 âŠ• 01000 = 11001 ( d ) (11011 âˆ¨ 01010) âˆ§ (10001 âˆ¨ 11011) = 11011 âˆ§ 11011 = 11011 . 1 Prob 1.2.30. We only need to examine the case when the conclusion q âˆ¨ r is false. This is possible if and only if q = F and r = F . But then p âˆ¨ q = F in case p = F , and Â¬ p âˆ¨ r = F in the other case p = T . So, regardless of the value of p , the assumption ( p âˆ¨ q ) âˆ§ ( Â¬ p âˆ¨ r ) is false if q âˆ¨ r is false. Hence this is a tautology. Remark: there are many other ways of solving this problem. Prob 1.3.10. (a) âˆƒ x ( C ( x ) âˆ§ D ( x ) âˆ§ F ( x )). (b) âˆ€ x ( C ( x ) âˆ¨ D ( x ) âˆ¨ F ( x )). (c) âˆƒ x ( C ( x ) âˆ§ F ( x ) âˆ§ Â¬ D ( x )). (d) Â¬âˆƒ x ( C ( x ) âˆ§ D ( x ) âˆ§ F ( x )). (e) âˆƒ xC ( x ) âˆ§ âˆƒ xD ( x ) âˆ§ âˆƒ xF ( x ). 2...
View Full Document

{[ snackBarMessage ]}

### Page1 / 2

hmw1 - b(01111 âˆ 10101 âˆ¨ 01000 = 00101 âˆ¨ 01000 = 01101...

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online