MAT 230 Module Two Homework

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.
1. Drink more water. No, it is not a statement; it is a command.
2. Paris is the capital city of the United States of America. Yes, it is a statement. It is false.
3. 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.

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

(p ∧ q)

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

(p v q)

1. The negation (~) of the conjunction (∧) of p and q. We cannot buy a book and we cannot go to a restaurant. (~p ∧ ~q)
2. 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): n2 = n + 6.
1. What is P(2) as a statement? ∀n ~P(2)
2. What is P(3) as a statement? ∃n P(3)
3. What is the truth value of ∀n P(n)? F
4. 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.

1. I will watch TV if I have finished my homework. p⇒q
2. I will watch TV only if I have not finished my homework. p⇒~q
3. I will watch TV is a necessary condition for I have finished my homework. p⇐q
4. I will not watch TV is a sufficient condition for I have finished my homework. ~p⇐q
5. 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.
1. Write the contrapositive of that statement. If Emily will not go to the museum, then it is not Friday. (~q ⇒~p)
2. 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.
1. 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

1. 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

1. 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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