Question 1

Consider the argument form: 
$p \wedge \sim q \rightarrow r$
$p \vee q$
$q \rightarrow p$
Therefore $r$. 
Use the truth table below to determine whether this argument form is valid or invalid. Annotate the table (as appropriate) and include a few words explaining how the truth table supports your answer.

Accepted Answer

The given form of argument is invalid.  Row 2 of the truth table shows that it is possible for an argument of this form to have true premises and a false conclusion.

Question 2

Write the following two statements in symbolic form and determine whether they are logically equivalent. Include a truth table and a few words explaining how the truth table supports your answer. 
If Sam bought it at Crown Books, then Sam didn't pay full price.
Sam bought it at Crown Books or Sam paid full price.

Accepted Answer

Solution 1: The given statements are not logically equivalent. Let p be "Sam bought it at Crown Books," and q be "Sam didn't pay full price." Then the two statements have the following form: $$p \rightarrow q \quad \text { and } \quad p \vee \sim q$$ The truth tables for these statement forms are $$\begin{array} { | l | l | l | | l | | l | } 
\hline p & q & \sim q & p \rightarrow q & p \vee \sim q \
\hline \hline T & T & F & T & T \
T & F & T & F & T \
F & T & F & T & F \
F & F & T & T & T \
\hline
\end{array}$$ Rows 2 and 3 of the table show that $p \rightarrow q$ and $p \vee \sim q$ do not always have the same truth values, and so $p \rightarrow q \not \neq p \vee \sim q$. 
Solution 2: The given statements are not logically equivalent. Let p be "Sam bought it at Crown
Books," and q be "Sam paid full price." Then the two statements have the following form: $$p \rightarrow \sim q \quad \text { and } \quad p \vee q$$ The truth tables for these statement forms are $$\begin{array} { | l | l | l | | l | | l | } 
\hline p & q & \sim q & p \rightarrow \sim q & p \vee q \
\hline \hline T & T & F & F & T \
T & F & T & T & T \
F & T & F & T & T \
F & F & T & T & F \
\hline
\end{array}$$ Rows 1 and 4 of the table show that $p \rightarrow \sim q$ and $p \vee q$ do not always have the same truth values, and so $p \rightarrow \sim q \not \equiv p \vee q$.

Question 3

Determine whether the following argument is valid or invalid. Include a truth table and a few words explaining why the truth table shows validity or invalidity.
If 12 divides 709,438 then 3 divides 709,438.
If the sum of the digits of 709,438 is divisible by 9 then 3 divides 709,438.
The sum of the digits of 709,438 is not divisible by 9.
Therefore, 12 does not divide 709,438.

Accepted Answer

Let p be "12 divides 709,438," q be "3 divides 709,438," and r be "The sum of the digits of 709,438 is
divisible by 9." Then the given argument has the following form: 
$p \rightarrow q$
$\begin{array} { l } 
r \rightarrow q \
\sim r
\end{array}$
Therefore $\sim p$.
 Row 2 of the truth table shows that it is possible for an argument of this form to have true premises and a false conclusion. Therefore, the given argument is invalid.

Question 4

Consider the argument form: 
$\begin{aligned}
& p \rightarrow \sim q \
& q \rightarrow \sim p \
\therefore \quad & p \vee q
\end{aligned}$ 
Use the truth table below to determine whether this form of argument is valid or invalid. Annotate the table (as appropriate) and include a few words explaining how the truth table supports your answer. $\begin{array} { | c c | | c | c | | c | c | c | } 
\hline p & q & \sim p & \sim q & p \rightarrow \sim q & q \rightarrow \sim p & p \vee q \
\hline T & T & F & F & F & F & T \
\hline T & F & F & T & T & T & T \
\hline F & T & T & F & T & T & T \
\hline F & F & T & T & T & T & F \
\hline
\end{array}$

Accepted Answer

The answer of Consider the argument form: 
\(\begin{aligned}
& p \rightarrow...

Question 5

Consider the following circuit.  
(a) Find the output of the circuit corresponding to the input P = 1, Q = 0, and R = 1.
(b) Write the Boolean expression corresponding to the circuit.

Accepted Answer

The answer of Consider the following circuit.  
(a) Find...

Question 6

Write a negation for each of the following statements:
(a) The variable S is undeclared and the data are out of order.
(b) The variable S is undeclared or the data are out of order.
(c) If Al was with Bob on the ﬁrst, then Al is innocent. 
(d) $- 5 < x < 2$ (where  x  is a particular real number)

Accepted Answer

The answer of Write a negation for each of the...

Question 7

Are the following statement forms logically equivalent: $p \vee q \rightarrow p$ and $p \vee ( \sim p \wedge q ) ?$ Include a truth table and a few words explaining how the truth table supports your answer.

Accepted Answer

The answer of Are the following statement forms logically equivalent:...

Question 8

Write the converse, inverse, and contrapositive of "If Ed is Sue's father, then Liu is Sue's cousin."

Accepted Answer

The answer of Write the converse, inverse, and contrapositive of...

Question 9

State precisely (but concisely) what it means for a form of argument to be valid.

Accepted Answer

The answer of State precisely (but concisely) what it means...

Question 10

State precisely (but concisely) what it means for two statement forms to be logically equivalent.

Accepted Answer

The answer of State precisely (but concisely) what it means...

Question 11

Write the form of the following argument. Is the argument valid or invalid? Justify your answer.
If Ann has the ﬂu, then Ann has a fever.
Ann has a fever.
Therefore, Ann has the ﬂu.

Accepted Answer

The answer of Write the form of the following argument....

Question 12

Which of the following is a negation for "Jim has grown or Joan has shrunk."

Accepted Answer

The negation of "Jim has grown or Joan has shrunk" is "Jim has not grown and Joan has not shrunk." This is because the negation of an "or" statement is an "and" statement with both parts negated.

Question 13

Write the converse, inverse, and contrapositive of "If Ann is Jan's mother, then Jose is Jan's cousin."

Accepted Answer

The answer of Write the converse, inverse, and contrapositive of...

Question 14

Write the following two statements in symbolic form and determine whether they are logically equivalent. Include a truth table and a few words explaining how the truth table supports your answer.
If Sam is out of Schlitz, then Sam is out of beer.
Sam is not out of beer or Sam is not out of Schlitz.

Accepted Answer

The answer of Write the following two statements in symbolic...

Question 15

Determine whether the following argument is valid or invalid. Include a truth table and a few words explaining why the truth table shows validity or invalidity.
If Hugo is a physics major or if Hugo is a math major, then he needs to take calculus.
Hugo needs to take calculus or Hugo is a math major.
Therefore, Hugo is a physics major or Hugo is a math major.

Accepted Answer

The answer of Determine whether the following argument is valid...

Question 16

Rewrite the following statement in if-then form without using the word "necessary": Getting an answer of 10 for problem 16 is a necessary condition for solving problem 16 correctly.

Accepted Answer

The answer of Rewrite the following statement in if-then form...

Question 17

Write the converse, inverse, and contrapositive of "If Al is Tom's cousin, then Jim is Tom's grandfather."

Accepted Answer

The answer of Write the converse, inverse, and contrapositive of...

Question 18

Which of the following is a negation for "Jim is inside and Jan is at the pool."

Accepted Answer

The negation of a statement "P and Q" is "not P or not Q." Therefore, the negation of "Jim is inside and Jan is at the pool" is "Jim is not inside or Jan is not at the pool."

Question 19

Write the form of the following argument. Is the argument valid or invalid? Justify your answer.
If 54,587 is a prime number, then 17 is not a divisor of 54,587.
17 is a divisor of 54,587. 
Therefore, 54,587 is not a prime number.

Accepted Answer

The answer of Write the form of the following argument....

Question 20

On the island of knights and knaves, you meet three natives, A, B, and C, who address you as follows:
A: At least one of us is a knave.
B: At most two of us are knaves.
What are A, B, and C?

Accepted Answer

The answer of On the island of knights and knaves,...

Quiz 2: The Logic of Compound Statements

Consider the following circuit. (a) Find the output of the circuit corresponding to the input P = 1, Q = 0, and R = 1. (b) Write the Boolean expression corresponding to the circuit.

Are the following statement forms logically equivalent: p∨q→pp \vee q \rightarrow pp∨q→p and p∨(∼p∧q)?p \vee ( \sim p \wedge q ) ?p∨(∼p∧q)? Include a truth table and a few words explaining how the truth table supports your answer.

Write the converse, inverse, and contrapositive of "If Ed is Sue's father, then Liu is Sue's cousin."

State precisely (but concisely) what it means for a form of argument to be valid.

State precisely (but concisely) what it means for two statement forms to be logically equivalent.

Write the form of the following argument. Is the argument valid or invalid? Justify your answer. If Ann has the ﬂu, then Ann has a fever. Ann has a fever. Therefore, Ann has the ﬂu.

Which of the following is a negation for "Jim has grown or Joan has shrunk."

Write the converse, inverse, and contrapositive of "If Ann is Jan's mother, then Jose is Jan's cousin."

Write the following two statements in symbolic form and determine whether they are logically equivalent. Include a truth table and a few words explaining how the truth table supports your answer. If Sam is out of Schlitz, then Sam is out of beer. Sam is not out of beer or Sam is not out of Schlitz.

Rewrite the following statement in if-then form without using the word "necessary": Getting an answer of 10 for problem 16 is a necessary condition for solving problem 16 correctly.

Write the converse, inverse, and contrapositive of "If Al is Tom's cousin, then Jim is Tom's grandfather."

Which of the following is a negation for "Jim is inside and Jan is at the pool."

Write the form of the following argument. Is the argument valid or invalid? Justify your answer. If 54,587 is a prime number, then 17 is not a divisor of 54,587. 17 is a divisor of 54,587. Therefore, 54,587 is not a prime number.

On the island of knights and knaves, you meet three natives, A, B, and C, who address you as follows: A: At least one of us is a knave. B: At most two of us are knaves. What are A, B, and C?

Are the following statement forms logically equivalent: $p \vee q \rightarrow p$ and $p \vee ( \sim p \wedge q ) ?$ Include a truth table and a few words explaining how the truth table supports your answer.