MAT 230 Module Two Homework General: Before beginning this homework, be sure to read the textbook sections and the material in Module Two. Type your solutions into this document and be sure to show all steps for arriving at your solution. Just giving a final number may not receive full credit. You may copy and paste mathematical symbols from the statements of the questions into your solution. This document was created using the Arial Unicode font. These homework problems are proprietary to SNHU COCE. They may not be posted on any nonSNHU website. The Institutional Release Statement in the course shell gives details about SNHU's use of systems that compare student submissions to a database of online, SNHU, and other universities' documents. SNHU MAT230 Page 1 of 5 Module Two Homework 1) State whether each of the following is a statement or is not a statement and explain why. If it is a statement, give its truth value. a) Drink more water. Is a declarative sentence (command) but not a statement. b) Paris is the capital city of the United States of America. Is a statement but it's false because Paris is not the capital city of United States. c) Is it going to rain tomorrow? This is a question not a statement. This problem is similar to Example 1 and to Exercise 1 in Section 2.1 of your SNHU MAT230 textbook. 2) Consider the two propositions. p: We can buy a book. q: We can go to a restaurant. Write each of the following statements in symbolic notation and as English sentences. a) The conjunction () of p and q. p q : We can buy a book and we can go to a restaurant. b) The disjunction () of p and q. p q : We can buy a book or we can go to a restaurant. c) The negation (~) of the conjunction () of p and q. ~ p ~q: We can not buy a book and we can not go to a restaurant. d) The negation (~) of the disjunction () of p and q. ~ p ~q: We can not buy a book or we can not go to a restaurant. This problem is similar to Examples 2-4 and to Exercises 5 and 10 in Section 2.1 of your SNHU MAT230 textbook. 3) Write the statement \"Every number is more than its reciprocal\" symbolically by first defining a predicate and then using a quantifier. This problem is similar to Example 8 and to Exercise 18 in Section 2.1 of your SNHU MAT230 textbook. P(X)= for every number x, x > 1/x 4) Let P(n): n2 = n + 6. a) What is P(2) as a statement? b) What is P(3) as a statement? c) What is the truth value of n P(n)? d) What is the truth value of n P(n)? This problem is similar to Examples 8 and 9 and to Exercises 19, 20, and 21 in Section 2.1 of your SNHU MAT230 textbook. SNHU MAT230 Page 2 of 5 Module Two Homework 5) Complete a truth table for (p ~q) (~p q) . There are multiple ways to set up the columns of a truth table, so you may need fewer or more columns than shown. p q p q ~q ~p ~ q ~ p (p ~ q) (~ p q T T T F F T T T F F T F F T F T T F T T T F F T T T T T This problem is similar to Example 5 and to Exercises 27-30 in Section 2.1 of your SNHU MAT230 textbook. 6) Use the following: p: I will watch TV. q: I have finished my homework. Write each of the following statements in terms of p, q, and logical connectives. a) I will watch TV if I have finished my homework. p q b) I will watch TV only if I have not finished my homework. q p c) I will watch TV is a necessary condition for I have finished my homework. q p d) I will not watch TV is a sufficient condition for I have finished my homework. ~ p q e) I will watch TV if and only if I have finished my homework. p q This problem is similar to Example 1 and to Exercises 1 and 2 in Section 2.2 of your SNHU MAT230 textbook. You may want to use the symbols , , or . 7) Consider the following statement: If it is Friday, then Emily will go to the museum. I will let p: If it is Friday and q: Emily will go to the museum a) Write the contrapositive of that statement. ~ q ~ p Emily won't go to the museum, then it is not Friday. b) Write the converse of that statement. q p Emily will go to the museum, then if it is Friday. This problem is similar to Example 2 and to Exercises 3 and 4 in Section 2.2 of your SNHU MAT230 textbook. SNHU MAT230 Page 3 of 5 Module Two Homework 8) Construct a truth table for (p q) (p q). Explain how this truth table shows whether this statement is a tautology, a contradiction (absurdity), or a contingency. This problem is similar to Example 5 and to Exercises 10-12 in Section 2.2 of your SNHU MAT230 textbook. p q p q ~q ~p ~ q ~ p (p q) (p q) T T T F F T T T F F T F F T F T F F T T T F F T T T T T This is a tautology??? Am I correct? 9) Write each of the arguments below symbolically and then explain whether it is valid or not. a) If it is hot outside, then I will go swimming. I will not go swimming. It is not hot outside. p q q p ____________ p q Both arguments are valid b) If it is not hot outside or if it is raining, then I will not go swimming. It is not raining. I will not go swimming. ~ q ~ p ~ p ~ q _______________ c) I will go swimming if and only if it is hot outside. I will not go swimming. It is not hot outside. This problem is similar to Examples 2-5 and to Exercises 1-9 in Section 2.3 of your SNHU MAT230 textbook. 10) Prove or disprove that if the product of two numbers (in ) is even, then at least one of them must be even. This problem is similar to Examples 8 and 9 and to Exercises 13-18 in Section 2.3 of your SNHU MAT230 textbook. SNHU MAT230 Page 4 of 5 Module Two Homework 11) Prove or disprove that if the sum of two numbers (in ) is even, then at least one of them must be even. This problem is similar to Examples 8 and 9 and to Exercises 23-26 in Section 2.3 of your SNHU MAT230 textbook. SNHU MAT230 Page 5 of 5 Module Two