**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 non-SNHU 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.

- 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.

- Drink more water.
*No, it is not a statement; it is a command.* - Paris is the capital city of the United States of America.
*Yes, it is a statement. It is false.* - Is it going to rain tomorrow?
*No, it is not a statement; it is a question.*

This problem is similar to Example 1 and to Exercise 1 in Section 2.1 of your SNHU MAT230 textbook.

- 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.

- The conjunction (∧) of p and q.
*We can buy a book and we can go to a restaurant.*

*(p ∧ q)*

- The disjunction (∨) of p and q.
*We can buy a book or we can go to a restaurant.*

*(p v q)*

- The negation (~) of the conjunction (∧) of p and q.
*We cannot buy a book and we cannot go to a restaurant. (~p ∧ ~q)* - The negation (~) of the disjunction (∨) of p and q.
*We cannot buy a book or we cannot go to a restaurant. (~p v ~q)*

This problem is similar to Examples 2–4 and to Exercises 5 and 10 in Section 2.1 of your SNHU MAT230 textbook.

- Write the statement “Every number is more than its reciprocal” symbolically by first defining a predicate and then using a quantifier.
*R(x) is the predicate; “more than it’s reciprocal”.*

*∀x R(x)*

This problem is similar to Example 8 and to Exercise 18 in Section 2.1 of your SNHU MAT230 textbook.

- Let P(n): n
^{2}= n + 6.

- What is P(2) as a statement?
*∀n ~P(2)* - What is P(3) as a statement?
*∃n P(3)* - What is the truth value of ∀n P(n)?
*F* - What is the truth value of ∃n P(n)?
*T*

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.

- 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 |
~p ∧ q |

T |
T |
F |
F |

T |
F |
T |
F |

F |
T |
F |
T |

F |
F |
F |
F |

This problem is similar to Example 5 and to Exercises 27–30 in Section 2.1 of your SNHU MAT230 textbook.

- 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.

- I will watch TV if I have finished my homework.
*p⇒q* - I will watch TV only if I have not finished my homework.
*p⇒~q* - I will watch TV is a necessary condition for I have finished my homework.
*p⇐q* - I will not watch TV is a sufficient condition for I have finished my homework.
*~p⇐q* - 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 ⇔.

- Consider the following statement: If it is Friday, then Emily will go to the museum.

- Write the contrapositive of that statement.
*If Emily will not go to the museum, then it is not Friday. (~q ⇒~p)* - Write the converse of that statement.
*If Emily will go to the museum, then it is Friday. (q⇒p)*

This problem is similar to Example 2 and to Exercises 3 and 4 in Section 2.2 of your SNHU MAT230 textbook.

- 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) |
⇒ |
(p ∨ q) |

T |
T |
T |
T |
T |

T |
F |
F |
F |
T |

F |
T |
F |
F |
T |

F |
F |
F |
F |
F |

*Contingency being that the result depends solely on the truth values of each variable.*

- Write each of the arguments below symbolically and then explain whether it is valid or not.

- If it is hot outside, then I will go swimming.
*p⇒q*

__I will not go swimming.__*~q*

∴ It is not hot outside.*~p*

*Not Valid*

- If it is not hot outside or if it is raining, then I will not go swimming.
*~p v q⇒~r*

__It is not raining.__*~q*

∴ I will not go swimming.*~r*

*Not Valid*

- I will go swimming if and only if it is hot outside.
*p⇔q*

__I will not go swimming.__*~p*

∴ It is not hot outside.*~q*

*Valid*

This problem is similar to Examples 2–5 and to Exercises 1–9 in Section 2.3 of your SNHU MAT230 textbook.

- 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 to Exercises 13–18 in Section 2.3 of your SNHU MAT230 textbook.

- 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.

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