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hmw1

Course: MATH 55, Spring 2009
School: Berkeley
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Word Count: 360

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to Solutions 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. Contrapositive: 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...

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to Solutions 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. Contrapositive: 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 T F pp F F p F T p p T T For parts (c) and (D) the table is below (columns five and six): p F F T T q F T F T p T T F F q T F T F p q T F F T p q F T 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 (f) statement is a contradiction. p F F T T q F T F T pq F T T F p q T F F T (p q) (p q) T T T T (p q) (p q) F F F F Prob 1.1.38. (a) (b) (c) (d) 11000 (01011 11011) = 11000 11011 = 11000 (01111 10101) 01000 = 00101 01000 = 01101 (01010 11011) 01000 = 10001 01000 = 11001 (11011 01010) (10001 11011) = 11011 11011 = 11011. 1 Prob 1.2.30. We only need to examine the case when the conclusion q r is false. This is possible if and only if q = F and r = F . But then p q = F in case p = F , and p r = F in the other case p = T . So, regardless of the value of p, the assumption (p q) (p r) is false if q r is false. Hence this is a tautology. Remark: there are many other ways of solving this problem. Prob 1.3.10. (a) x (C(x) D(x) F (x)). (b) x (C(x) D(x) F (x)). (c) x (C(x) F (x) D(x)). (d) x (C(x) D(x) F (x)). (e) xC(x) xD(x) xF (x). 2
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