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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 ℕ) is even,​ then at least one of them must be even. This problem is similar to Examples 8 and 9 and toExercises 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 ℕ) 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 Homework

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