Week 1 Discrete Mathematics

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

32

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

4

Determine whether the given conditional statements is true or false. If 1 + 1 = 2, then dogs can fly

False

Write the given sentence in the "if p, then q" form. It is necessary to have a valid password to log on to the server.

If you log on to the server, then you have a valid password.

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

64

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

8

The statement is related to inhabitants of the island of knights and knaves created by Smullyan, where knights always tell the truth and knaves always lie. You encounter two people, A and B. Determine what A and B are if they address you as: A says "The two of us are both knights" and B says "A is a knave."

A is a knave and B is a knight.

State the converse, contrapositive, and inverse of this conditional statement. Click and drag the named related conditionals to their corresponding statements of the conditional statement "If it snows today, I will ski tomorrow."

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

Find the dual of the given compound proposition. The dual of the compound proposition p ∧ (q ∨ (r ∧ T)) is p∨∨(q ∧ (r ∨ T)).

False

Let p and q be the propositionsp: 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

Let p, q, and r be the propositionsp: You have the flu.q: You miss the final examination.r: You pass the course. NOTE: This is a multi-part question. Once an answer is submitted, you will be unable to return to this part. An English translation of the compound propositionq→ ¬ris "If you miss the final exam, then you pass the course."

False

Let p, q, and r be the propositionsp: You have the flu.q: You miss the final examination.r: You pass the course. Identify the English translation of the compound proposition (p → ¬r) ∨ (q → ¬r).

If you have the flu then you will not pass the course, or if you miss the final exam then you will not pass the course.

Let p, q, and r be the propositionsp: 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.

When planning a party you want to know whom to invite. Among the people you would like to invite are three touchy friends. You know that if Jasmine attends, she will become unhappy if Samir is there, Samir will attend only if Kanti will be there, and Kanti will not attend unless Jasmine also does. Which combinations of these three friends can you invite so as not to make someone unhappy? (Check all that apply.)

Jasmine and Kanti Jamine

Consider the following propositions p and q. p: "Swimming at the New Jersey shore is allowed." q: "Sharks have been spotted near the shore." Can the compound proposition p → ¬q be expressed as "If swimming at the New Jersey shore is not allowed, then sharks have been spotted near the shore"?

No

Determine whether the given compound propositions is satisfiable. (p ∨ q ∨ r) ∧ (p ∨ ¬q ∨ ¬s) ∧ (q ∨ ¬r ∨ s) ∧ (¬p ∨ r ∨ s) ∧ (¬p ∨ q ∨ ¬s) ∧ (p ∨ ¬q ∨ ¬r) ∧ (¬p ∨ ¬q ∨ s) ∧ (¬p ∨ ¬r ∨ ¬s) The compound proposition (p ∨ q ∨ r) ∧ (p ∨ ¬q ∨ ¬s) ∧ (q ∨ ¬r ∨ s) ∧ (¬p ∨ r ∨ s) ∧ (¬p ∨ q ∨ ¬s) ∧ (p ∨ ¬q ∨ ¬r) ∧ (¬p ∨ ¬q ∨ s) ∧ (¬p ∨ ¬r ∨ ¬s) is

Satisfiable

Determine whether the given compound propositions is satisfiable. (p ∨ q ∨ ¬r) ∧ (p ∨ ¬q ∨ ¬s) ∧ (p ∨ ¬r ∨ ¬s) ∧ (¬p ∨ ¬q ∨ ¬s) ∧ (p ∨ q ∨ ¬s) The compound proposition (p ∨ q ∨ ¬r) ∧ (p ∨ ¬q ∨ ¬s) ∧ (p ∨ ¬r ∨ ¬s) ∧ (¬p ∨ ¬q ∨ ¬s) ∧ (p ∨ q ∨ ¬s) is

Satisfiable

Determine whether the given compound propositions is satisfiable. (¬p ∨ ¬q ∨ r) ∧ (¬p ∨ q ∨ ¬s) ∧ (p ∨ ¬q ∨ ¬s) ∧ (¬p ∨ ¬r ∨ ¬s) ∧ (p ∨ q ∨ ¬r) ∧ (p ∨ ¬r ∨ ¬s) The compound proposition (¬p ∨ ¬q ∨ r) ∧ (¬p ∨ q ∨ ¬s) ∧ (p ∨ ¬q ∨ ¬s) ∧ (¬p ∨ ¬r ∨ ¬s) ∧ (p ∨ q ∨ ¬r) ∧ (p ∨ ¬r ∨ ¬s) is

Satisfiable

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

Swimming at the New Jersey shore is allowed, and sharks have been spotted near the shore.

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

Swimming at the New Jersey shore is allowed, or sharks have not been spotted near the shore.

Show that each of these conditional statements is a tautology by using truth tables. Complete the truth table given below for the conditional statement [(p ∨∨ q) ∧∧ (p → r) ∧∧ (q → r)] → r.

Tautology: True

Consider the following propositions p and q. p: "Swimming at the New Jersey shore is allowed." q: "Sharks have been spotted near the shore." The compound proposition p ↔ ¬q is "Swimming at the New Jersey

The compound proposition p ↔ ¬q is "Swimming at the New Jersey shore is allowed if and only if sharks have not been spotted near the shore."

Which of these are propositions? What are the truth values of those that are propositions? The moon is made of green cheese.

The given statement is a proposition , and the truth value is false

Complete the truth table given below for the statement (¬p ∧ (p → q)) → ¬q.

The given statement is a: Contingency

Which of these are propositions? What are the truth values of those that are propositions? 2^n ≥100

The given statement is not a proposition , and the truth value does not exist

Which of these are propositions? What are the truth values of those that are propositions? 4 + x = 5

The given statement is not a proposition , and the truth value does not exist

Which of these are propositions? What are the truth values of those that are propositions? Do not pass go. The given statement is __________________ , and the truth value __________________

The given statement is not a proposition , and the truth value does not exist

Which of these are propositions? What are the truth values of those that are propositions? What time is it?

The given statement is not a proposition, and the truth value does not exist

Let p and q be the propositionsp: It is below freezing.q: It is snowing. The compound proposition p → q is "If it is below freezing, it is also snowing."

The proposition for the statement "It is not below freezing" is ¬p. The proposition for the statement "It is below freezing" is p. The proposition for the statement "It is not snowing" is ¬q. The proposition for the statement "It is not below freezing and it is not snowing" is ¬p ∧ ¬q.

Which of these are propositions? What are the truth values of those that are propositions? There are no black flies in Maine

There are no black flies in Maine. The given statement is a proposition , and the truth value is false

Consider the following propositions p and q. p: "Swimming at the New Jersey shore is allowed." q: "Sharks have been spotted near the shore." Check if the proposition "Sharks have not been spotted near the shore" is the expression for ¬q.

True

Determine whether the given conditional statements is true or false. If 1 + 1 = 3, then dogs can fly.

True

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

True

Determine whether the given conditional statements is true or false. If 2 + 2 = 4, then 1 + 2 = 3.

True

Find the dual of the given compound proposition. The dual of the compound position (p∧¬q)∨(q∧F)(p∧¬q)∨(q∧F) is (p∨¬q)∧(q∨T)(p∨¬q)∧(q∨T) .

True

Let p and q be the propositionsp: It is below freezing.q: It is snowing. The compound proposition for the statement "It is below freezing and snowing" is p ∧ q.

True

Let p, q, and r be the propositionsp: You have the flu.q: You miss the final examination.r: You pass the course. An English translation of the compound proposition (p ∧ q) ⊕ (¬q ∧ r) is "Either you have the flu and miss the final exam, or you do not miss the final exam and do pass the course."

Yes

Let p, q, and r be the propositionsp: You have the flu.q: You miss the final examination.r: You pass the course. An English translation of the compound proposition ¬q ↔ r is "You do not miss the final exam if and only if you pass the course."

Yes

Let p, q, and r be the propositionsp: 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.

You have the flu, or miss the final examination, or pass the course.

When planning a party you want to know whom to invite. Among the people you would like to invite are three touchy friends. You know that if Jasmine attends, she will become unhappy if Samir is there, Samir will attend only if Kanti will be there, and Kanti will not attend unless Jasmine also does. Let j, s, and k denote the propositions that Jasmine, Samir, and Kanti attend, respectively. Express the given conditions using logical connectives.

j → ¬s, s → k, and ¬k ∨ j

Let p, q, and r be the propositionsp: 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 on the final, you do every exercise in this book, and you get an A in this class" using p, q, and r and logical connectives (including negations).

p ∧ q ∧ r

Find the dual of the given compound proposition. p ∨ ¬q

p ∧ ¬q

Let p and q be the propositionsp: It is below freezing.q: It is snowing. The compound proposition for the statement "It is below freezing but not snowing" is _____.

p ∧ ¬q

Let p, q, and r be the propositionsp: 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 on the final, but you don't do every exercise in this book; nevertheless, you get an A in this class" using p, q, and r and logical connectives (including negations).

p ∧ ¬q ∧ r

Let p, q, and r be the propositionsp: 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 "To get an A in this class, it is necessary for you to get an A on the final" using p, q, and r and logical connectives (including negations).

r → p

Let p, q, and r be the propositionsp: 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