Chapter 1 Section 1.1

Complete the truth table given below for the given compound proposition. p | ¬p | p v ¬p T F F T

p v ¬p T T

p: You drive over 65 miles per hour. q: You get a speeding ticket. Identify the expression that represents the proposition "Driving over 65 miles per hour is sufficient for getting a speeding ticket" using p and q and logical connectives.

p → q. In the given statement, p is the hypothesis and q is the conclusion. The proposition of the given statement is p → q.

Complete the truth table given below for the given proposition. p → ¬p.

p → ¬p. F T

Complete the truth table given below for the given compound proposition. p | ¬p | p ∧ ¬p T F T T

p ∧ ¬p F F

Complete the truth table given below for the given compound proposition. (p ∨ ¬q) → q.

(p ∨ ¬q) → q. T F T F

How many rows appear in a truth table for this compound propositions? (p∨¬t)∧(p∨¬s)(p∨¬t)∧(p∨¬s) The number of rows needed for the truth table of the compound proposition (p∨¬t)∧(p∨¬s)(p∨¬t)∧(p∨¬s) is ____.

8 A truth table will need 2n rows if there are n variables. In this case, three variables —p, t, and s; are used.

State the converse, contrapositive, and inverse of this conditional statement.

Converse: If i am to ski tomorrow, it must snow today. Inverse: If it does not snow today, then I will not ski tomorrow. Contrapositive: If I do not ski tomorrow, then it will not have snowed today.

Determine whether the given sentence is an inclusive or, or an exclusive or, is intended. Lunch includes soup or salad.

Exclusive or The restaurant would probably charge extra if the diner wanted both of these items, so this is an exclusive "or".

p: "Swimming at the New Jersey shore is allowed." q: "Sharks have been spotted near the shore." The compound proposition ¬p → ¬q is "If sharks have not been spotted near the shore, then swimming at the New Jersey shore is not allowed."

False The compound proposition in the statement represents ¬q → ¬p.

p: It is below freezing. q: It is snowing. The compound proposition for the statement "It is either snowing or below freezing (or both)" is p ∧ q.

False The compound proposition is p V q

2 + 1 = 3. The negation of the given statement is "2 × 1 ≠ 3".

False The negation of P: 2 + 1 = 3, is "2 + 1 ≠ 3".

Mei has an MP3 player.

Identify the negations of the given proposition. (Check all that apply.) it is not the case that Mei has an MP3 player. Mei has no MP3 player.

p: You have the flu. q: You miss the final examination. r: You pass the course. Identify an English translation that expresses the compound proposition p → q.

If you have the flu, then you miss the final examination. The compound proposition p → q of the hypothesis p and the conclusion q is translated in English as "if p, then q."

Determine whether the given sentence is an inclusive or, or an exclusive or, is intended. To enter the country you need a passport or a voter registration card.

Inclusive or If a person happened to have both forms of identification, so much the better, so this is clearly an inclusive "or".

Identify the negation of "Jennifer and Teja are friends."

Jennifer and Teja are not friends.

p: You have the flu. q: You miss the final examination. r: You pass the course. Identify the statement that express the compound proposition p ∨ q ∨ r as an English sentence.

The "vel" symbol represents the word "or". So, the English statement of the given proposition is "You have the flu, or miss the final examination, or pass the course."

p: It is below freezing. q: It is snowing. Identify the steps that are used to find the compound proposition for the statement "Either it is below freezing or it is snowing, but it is not snowing if it is below freezing."

The compound proposition for the statement "Either it is below freezing or it is snowing, but it is not snowing if it is below freezing" is (p ∨ q) ∧ (p → ¬q).

Do not pass go.

The given statement is: Not a proposition because it is a command, and the truth value does not exist

4 + x = 5

The given statement is: Not a proposition, and the truth value does not exist because it depends on the value of x.

Identify the compound proposition p ∧ q. p: "Swimming at the New Jersey shore is allowed." q: "Sharks have been spotted near the shore."

The symbol ∧ denotes the word 'and'. So, the compound proposition p ∧ q is "Swimming at the New Jersey shore is allowed, and sharks have been spotted near the shore."

Determine whether the biconditional is true or false. 1 + 1 = 3 if and only if monkeys can fly.

True The given biconditional statement is F ↔ F, which is true.

Determine whether the given conditional statements is true or false. If 1 + 1 = 3, then unicorns exist.

True This is F → F, which is true.

Identify the propositions in the form "p if and only if q" in English. If it is hot outside you buy an ice cream cone, and if you buy an ice cream cone it is hot outside. (Check all that apply.)

You buy an ice cream cone if and only if it is hot outside. You do not buy an ice cream cone if and only if it is not hot outside. The "p if and only if q" form of the biconditional statement "If it is hot outside you buy an ice cream cone, and if you buy an ice cream cone it is hot outside" are "You buy an ice cream cone if and only if it is hot outside," and "You do not buy an ice cream cone if and only if it is not hot outside."

p: You get an A on the final exam. q: You do every exercise in this book. r: You get an A in this class. Identify the expression for the proposition "You get an A in this class, but you do not do every exercise in this book" using p, q, and r and logical connectives (including negations).

r ∧ ¬q The proposition for the statement "you do not do every exercise in this book" is ¬q. Hence the compound proposition for the statement "You get an A in this class, but you do not do every exercise in this book" is r ∧ ¬q.

p: You drive over 65 miles per hour. q: You get a speeding ticket. Identify the expression that represents the proposition "You do not drive over 65 miles per hour" using p and logical connectives (including negations).

¬p