<p> CmSc175 Discrete Mathematics Homework 04 due 02/11 </p><p>1. Find the contrapositive, converse, inverse, and the disjunctive form of the conditional</p><p>~P V ~ Q  P V ~R </p><p>Apply DeMorgan’s laws, and absorption laws where applicable.</p><p>Contrapositive:</p><p>Converse:</p><p>Inverse:</p><p>Disjunctive form:</p><p>2. Using the equivalence laws to show that ~( (~P V ~ Q)  (P V ~R) ) = ~P  R </p><p>At each step indicate the law being used.</p><p>3. Rephrase the following sentences as “if – then” sentences: </p><p> a. Sam will be allowed on Tom’s racing boat only if he is an expert sailor.</p><p> b. This door will not open unless a security code is entered</p><p> c. Having two 45 angles is a sufficient condition for this triangle to be a right triangle </p><p> d. A necessary condition for this compound to boil is that its temperature be at least 250 F</p><p>1 4. In the following problem, a set of premises and a conclusion are given. Show how the conclusion can be deduced from the premises. At each step write the inference rule used.</p><p>(1) P V Q (2) Q  R</p><p>(3) P  S  Z (4) ~R</p><p>(5) ~Q  U  S Therefore Z For example, the first step in the proof would be: (6) ~Q by (2), (4) and Modus Tollens</p><p>5. Using the predicates healthy(x), wealthy(x), happy (x), wise(x) and appropriate quantifiers (,), represent as predicate statements the following sentences and write the negation of the quantified expressions: a. All wealthy people are happy Expression: Negation: b. Some healthy people are happy Expression: Negation: c. Some wise people are happy and wealthy Expression: Negation:</p><p> d. All happy and wealthy people are wise</p><p>2 Expression: Negation:</p><p>6. Indicate whether the following arguments in predicate logic are valid or invalid (underline the correct answer). If valid, indicate the inference rule. If invalid, indicate the type of the error (if inverse or converse) Note: Some errors do not have names, for example: All students study Peter likes cherries Therefore Peter is a student This is an invalid argument, the error does not have a name.</p><p> a. All Math contest participants are CS students or Math majors Peter is not a participant in the Math contest Therefore Peter is not a CS student</p><p> b. All Math contest participants are double majors CS and Math Peter is not a Math major Therefore Peter is not a participant in the Math contest</p><p> c. All Math contest participants are CS students or Math majors Peter is not a Math major Therefore Peter is not a participant in the Math contest</p><p>7. Give direct proof of the following statement: For all n, if n is odd then n2 is odd</p><p>8. Consider the sequence 1, 3, 7, 15, 31, ….given recursively:</p><p> a0 = 1 an+1 = 2*an + 1 (* is multiplication)</p><p>Using mathematical induction prove the following statement n + 1 P(n): an = 2 – 1, where n = 0, 1, 2, …</p><p>3</p>