Question 1

Restate in a logically equivalent form.
-The flowers are not blooming or it is not winter.
A)It is not true that the flowers are blooming and it is winter.
B)The flowers are blooming and it is winter.
C)It is not true that both the flowers are blooming and it is winter.
D)It is not true that it is winter and the flowers are not blooming.

Accepted Answer

The answer of Restate in a logically equivalent form.
-The flowers...

Question 2

Construct a truth table for the statement.
-$ (t \wedge p) \vee(\sim t \wedge \sim p) $
A)
$\begin{array}{llc}
\mathrm{t} & \mathrm{p} & (\mathrm{t} \wedge \mathrm{p}) \vee(\sim \mathrm{t} \wedge \sim \mathrm{p}) \
\hline \mathrm{T} & \mathrm{T} & \mathrm{T} \
\mathrm{T} & \mathrm{F} & \mathrm{F} \
\mathrm{F} & \mathrm{T} & \mathrm{F} \
\mathrm{F} & \mathrm{F} & \mathrm{T}
\end{array}$

B)

C)
$\begin{array}{llc}
t & \mathrm{p} & (t \wedge p) \vee(\sim t \wedge \sim \mathrm{p}) \
\hline \mathrm{T} & \mathrm{T} & \mathrm{T} \
\mathrm{T} & \mathrm{F} & \mathrm{T} \
\mathrm{F} & \mathrm{T} & \mathrm{T} \
\mathrm{F} & \mathrm{F} & \mathrm{F}
\end{array}$

D)
$\begin{array}{lll}
t & \mathrm{p} & (\mathrm{t} \wedge \mathrm{p}) \vee(\sim t \wedge \sim \mathrm{p}) \
\hline \mathrm{T} & \mathrm{T} & \mathrm{F} \
\mathrm{T} & \mathrm{F} & \mathrm{F} \
\mathrm{F} & \mathrm{T} & \mathrm{T} \
\mathrm{F} & \mathrm{F} & \mathrm{T}
\end{array}$

Accepted Answer

The answer of Construct a truth table for the statement.
-\(...

Question 3

Letting r stand for "The food is good," p stand for "I eat too much," and q stand for "I'll exercise," write the following in symbolic form.
-If I eat too much, then I'll exercise. 
A) $p \rightarrow q$
B) $r \rightarrow p$
C) $\mathrm { q } \rightarrow \mathrm { p }$
D) $p \vee q$

Accepted Answer

The statement "If I eat too much, then I'll exercise" directly translates to $p \rightarrow q$, where $p$ represents "I eat too much" and $q$ represents "I'll exercise."

Question 4

Letting r stand for "The food is good," p stand for "I eat too much," and q stand for "I'll exercise," write the following in symbolic form.
-I'll exercise if I don't eat too much. 
A) $\sim p \rightarrow q$
B) $\sim p \wedge q$
C) $\sim ( p \rightarrow q )$
D) $\sim p \vee q$

Accepted Answer

The statement "I'll exercise if I don't eat too much" can be translated to "If not p, then q," which is symbolically represented as $\sim p \rightarrow q$.

Question 5

Construct a truth table for the statement.
-$\sim(\mathrm{s} \vee \mathrm{t}) \wedge \sim(\mathrm{t} \wedge \mathrm{s})$
A)
$\begin{array}{ccc}
\mathrm{s} & \mathrm{t} & \sim(\mathrm{s} \vee \mathrm{t}) \wedge \sim(\mathrm{t} \wedge \mathrm{s}) \
\hline \mathrm{T} & \mathrm{T} & \mathrm{F} \
\mathrm{T} & \mathrm{F} & \mathrm{F} \
\mathrm{F} & \mathrm{T} & \mathrm{F} \
\mathrm{F} & \mathrm{F} & \mathrm{T}
\end{array}$

B)
$\begin{array}{ccc}
\mathrm{s} & \mathrm{t} & \sim(\mathrm{s} \vee \mathrm{t}) \wedge \sim(\mathrm{t} \wedge \mathrm{s}) \
\hline \mathrm{T} & \mathrm{T} & \mathrm{F} \
\mathrm{T} & \mathrm{F} & \mathrm{F} \
\mathrm{F} & \mathrm{T} & \mathrm{F} \
\mathrm{F} & \mathrm{F} & \mathrm{F}
\end{array}$

C)

D)
$\begin{array}{lll}
\mathrm{s} & \mathrm{t} & \sim(\mathrm{s} \vee \mathrm{t}) \wedge \sim(\mathrm{t} \wedge \mathrm{s}) \
\hline \mathrm{T} & \mathrm{T} & \mathrm{F} \
\mathrm{T} & \mathrm{F} & \mathrm{T} \
\mathrm{F} & \mathrm{T} & \mathrm{T} \
\mathrm{F} & \mathrm{F} & \mathrm{F}
\end{array}$

Accepted Answer

The answer of Construct a truth table for the statement.
-\(\sim(\mathrm{s}...

Question 6

Letting r stand for "The food is good," p stand for "I eat too much," and q stand for "I'll exercise," write the following in symbolic form.
-If I exercise, then I won't eat too much. 
A) $\sim ( \mathrm { p } \rightarrow \mathrm { q } )$
B) $r \wedge p$
C) $p \rightarrow q$
D) $q \rightarrow \sim p$

Accepted Answer

The statement "If I exercise, then I won't eat too much" directly translates to "If q, then not p," which is symbolically represented as $q \rightarrow \sim p$.

Question 7

Construct a truth table for the statement.
-$\sim ( \sim ( \mathrm { s } \vee \mathrm { p } ) )$
A)
$\begin{array}{ccc}
\mathrm{s} & \mathrm{p} & \sim(\sim(\mathrm{s} \vee \mathrm{p})) \
\hline \mathrm{T} & \mathrm{T} & \mathrm{T} \
\mathrm{T} & \mathrm{F} & \mathrm{T} \
\mathrm{F} & \mathrm{T} & \mathrm{F} \
\mathrm{F} & \mathrm{F} & \mathrm{F}
\end{array}$

B) 
$\begin{array}{ccc}
\mathrm{s} & \mathrm{p} & \sim(\sim(\mathrm{s} \mathrm{Vp})) \
\hline \mathrm{T} & \mathrm{T} & \mathrm{F} \
\mathrm{T} & \mathrm{F} & \mathrm{F} \
\mathrm{F} & \mathrm{T} & \mathrm{F} \
\mathrm{F} & \mathrm{F} & \mathrm{T}
\end{array}$

C)
$\begin{array}{ccc}
\mathrm{s} & \mathrm{p} & \sim(\sim(\mathrm{s} \mathrm{} \vee \mathrm{p})) \
\hline \mathrm{T} & \mathrm{T} & \mathrm{T} \
\mathrm{T} & \mathrm{F} & \mathrm{T} \
\mathrm{F} & \mathrm{T} & \mathrm{T} \
\mathrm{F} & \mathrm{F} & \mathrm{F}
\end{array}$

D) 
$\begin{array}{ccc}
\mathrm{s} & \mathrm{p} & \sim(\sim(\mathrm{s} \vee \mathrm{p})) \
\hline \mathrm{T} & \mathrm{F} & \mathrm{T} \
\mathrm{F} & \mathrm{T} & \mathrm{F}
\end{array}$

Accepted Answer

The statement $\sim ( \sim ( \mathrm { s } \vee \mathrm { p } ) )$ is a double negation of the disjunction of $s$ and $p$. The double negation cancels out, leaving the truth value of $s \vee p$. Thus, the statement is true whenever $s$ or $p$ (or both) is true, and false only when both are false.

Question 8

Restate in a logically equivalent form.
-It is not true that both this book is interesting and the book is about stars.
A)This book cannot be both interesting and about stars.
B)Either this book is not interesting or it is not about stars.
C)This book is both interesting and about stars.
D)Either this book is interesting or it is about stars.

Accepted Answer

The original statement negates the conjunction of two propositions: the book being interesting and about stars. The logically equivalent form is to apply De Morgan's Law, which turns the conjunction into a disjunction with each proposition negated. This results in the statement that either the book is not interesting or it is not about stars.

Question 9

Restate in a logically equivalent form.
-It is not true that today I both went to school and read a book.
A)Today, I did not read a book and did not go to school.
B)Today, I read a book but did not go to school.
C)Today, I either did not go to school or I did not read a book.
D)Today, I went to school and read a book.

Accepted Answer

The original statement negates the conjunction of two actions (going to school and reading a book), which logically translates to at least one of the actions not happening. This is best represented by an "either...or" statement where at least one of the negated actions must be true, making option C the logically equivalent form.

Question 10

Letting r stand for "The food is good," p stand for "I eat too much," and q stand for "I'll exercise," write the following in symbolic form.
-If I exercise, then the food won't be good and I won't eat too much. 
A) $\sim ( r \wedge p ) \rightarrow q$
B) $q \rightarrow ( \sim r \wedge \sim p )$
C) $q \rightarrow \sim ( r \wedge p )$
D) $( \mathrm { q } \wedge \sim \mathrm { r } ) \rightarrow \sim \mathrm { p }$

Accepted Answer

The statement "If I exercise, then the food won't be good and I won't eat too much" directly translates to $q \rightarrow ( \sim r \wedge \sim p )$, where $q$ is "I'll exercise", $\sim r$ is "The food won't be good", and $\sim p$ is "I won't eat too much".

Question 11

Letting r stand for "The food is good," p stand for "I eat too much," and q stand for "I'll exercise," write the following in symbolic form.
-If the food is not good, I won't eat too much. 
A) $\sim p \rightarrow \sim r$
B) $\sim ( \mathrm { r } - \mathrm { p } )$
C) $\sim r \rightarrow \sim p$
D) $\mathrm { r } \rightarrow \sim \mathrm { p }$

Accepted Answer

The statement "If the food is not good, I won't eat too much" translates to "If not r, then not p," which is symbolically represented as $\sim r \rightarrow \sim p$.

Question 12

Letting r stand for "The food is good," p stand for "I eat too much," and q stand for "I'll exercise," write the following in symbolic form.
-If the food is good or if I eat too much, I'll exercise. 
A) $r \rightarrow p \rightarrow r$
B) $( r \vee p ) \rightarrow q$
C) $r \rightarrow ( p \vee q )$
D) $( r \wedge p ) \rightarrow q$

Accepted Answer

The correct symbolic form is $( r \vee p ) \rightarrow q$, which translates to "If the food is good or if I eat too much, then I'll exercise." This captures the essence of the statement by using the logical OR ($\vee$) to combine the two conditions (good food or eating too much) as a single premise leading to the conclusion (exercising).

Question 13

Construct a truth table for the statement.
-$(p \wedge w) \wedge(\sim w \vee t)$
A)
$\begin{array}{cccc}
\mathrm{p} & \mathrm{w} & \mathrm{t} & (\mathrm{p} \wedge \mathrm{W}) \wedge(\sim \mathrm{W} \vee \mathrm{t}) \
\hline \mathrm{T} & \mathrm{T} & \mathrm{T} & \mathrm{T} \
\mathrm{T} & \mathrm{T} & \mathrm{F} & \mathrm{F} \
\mathrm{T} & \mathrm{F} & \mathrm{T} & \mathrm{F} \
\mathrm{T} & \mathrm{F} & \mathrm{F} & \mathrm{F} \
\mathrm{F} & \mathrm{T} & \mathrm{T} & \mathrm{F} \
\mathrm{F} & \mathrm{T} & \mathrm{F} & \mathrm{F} \
\mathrm{F} & \mathrm{F} & \mathrm{T} & \mathrm{F} \
\mathrm{F} & \mathrm{F} & \mathrm{F} & \mathrm{F}
\end{array}$

B)
$\begin{array}{cccc}
\mathrm{p} & \mathrm{w} & \mathrm{t} & (\mathrm{p} \wedge \mathrm{w}) \wedge(\sim \mathrm{w} \vee \mathrm{t}) \
\hline \mathrm{T} & \mathrm{T} & \mathrm{T} & \mathrm{F} \
\mathrm{T} & \mathrm{T} & \mathrm{F} & \mathrm{T} \
\mathrm{T} & \mathrm{F} & \mathrm{T} & \mathrm{T} \
\mathrm{T} & \mathrm{F} & \mathrm{F} & \mathrm{T} \
\mathrm{F} & \mathrm{T} & \mathrm{T} & \mathrm{T} \
\mathrm{F} & \mathrm{T} & \mathrm{F} & \mathrm{F} \
\mathrm{F} & \mathrm{F} & \mathrm{T} & \mathrm{T} \
\mathrm{F} & \mathrm{F} & \mathrm{F} & \mathrm{T}
\end{array}$

Accepted Answer

The correct truth table reflects the logical operations accurately. For the expression $(p \wedge w) \wedge(\sim w \vee t)$, the result is true only when both $p$ and $w$ are true, and either $w$ is false or $t$ is true. However, since $w$ cannot be both true and false simultaneously, the only time the entire expression is true is when $p$, $w$, and $t$ are all true.

Question 14

Letting r stand for "The food is good," p stand for "I eat too much," and q stand for "I'll exercise," write the following in symbolic form.
-If the food is good and if I eat too much, then I'll exercise. 
A) $r \wedge ( p \rightarrow q )$
B) $( r \wedge p ) \rightarrow q$
C) $p \rightarrow ( r \wedge q )$
D) $r \rightarrow ( p \wedge q )$

Accepted Answer

The statement "If the food is good and if I eat too much, then I'll exercise" directly translates to the logical expression $( r \wedge p ) \rightarrow q$, indicating that if both conditions $r$ and $p$ are true, then $q$ will also be true.

Question 15

Construct a truth table for the statement.
-$w \vee ( w \wedge \sim w )$
A)
$\begin{array}{ll}
\mathrm{W} & \mathrm{W} \vee(\mathrm{w} \wedge \sim \mathrm{w}) \
\hline \mathrm{T} & \mathrm{F} \
\mathrm{F} & \mathrm{F}
\end{array}$

B)
$\begin{array}{lc}
\mathrm{W} & \mathrm{W} \vee(\mathrm{w} \wedge \sim \mathrm{W}) \
\hline \mathrm{T} & \mathrm{T} \
\mathrm{F} & \mathrm{F}
\end{array}$

C)
$\begin{array}{lc}
\mathrm{W} & \mathrm{W} \vee(\mathrm{W} \wedge \sim \mathrm{W}) \
\hline \mathrm{T} & \mathrm{T} \
\mathrm{F} & \mathrm{T}
\end{array}$

D)
$\begin{array}{lc}
\mathrm{W} & \mathrm{W} \vee(\mathrm{W} \wedge \sim \mathrm{W}) \
\mathrm{T} & \mathrm{F} \
\mathrm{F} & \mathrm{T}
\end{array}$

Accepted Answer

The expression $w \wedge \sim w$ is always false because $w$ and $\sim w$ (not $w$) are opposites and cannot both be true at the same time. Therefore, $w \vee (w \wedge \sim w)$ simplifies to $w \vee F$, which is just $w$. So, if $w$ is true, the whole expression is true; if $w$ is false, the whole expression is false.

Question 16

Restate in a logically equivalent form.
-If a number is divisible by 4 , then it is divisible by 2 .
A) If a number is not divisible by 4 , then it is not divisible by $2 .$
B) If a number is not divisible by 2 , then it is not divisible by 4 .
C) If a number is divisible by 4 , then it is not divisible by $2 .$
D) If a number is not divisible by 4 , then it is divisible by $2 .$

Accepted Answer

The correct restatement using logical equivalence involves contraposition, which flips and negates both the hypothesis and the conclusion. Thus, "If a number is not divisible by 2, then it is not divisible by 4" is logically equivalent to the original statement.

Question 17

Construct a truth table for the statement.
-$\sim((w \wedge \mathrm{q}) \vee \mathrm{S})$
A)
$\begin{array}{cccc}
\mathrm{w} & \mathrm{q} & \mathrm{s} & \sim((\mathrm{w} \wedge \mathrm{q}) \vee \mathrm{s}) \
\hline \mathrm{T} & \mathrm{T} & \mathrm{T} & \mathrm{T} \
\mathrm{T} & \mathrm{T} & \mathrm{F} & \mathrm{F} \
\mathrm{T} & \mathrm{F} & \mathrm{T} & \mathrm{T} \
\mathrm{T} & \mathrm{F} & \mathrm{F} & \mathrm{F} \
\mathrm{F} & \mathrm{T} & \mathrm{T} & \mathrm{T} \
\mathrm{F} & \mathrm{T} & \mathrm{F} & \mathrm{F} \
\mathrm{F} & \mathrm{F} & \mathrm{T} & \mathrm{T} \
\mathrm{F} & \mathrm{F} & \mathrm{F} & \mathrm{F}
\end{array}$
B)$\begin{array}{cccc}
\mathrm{w} & \mathrm{q} & \mathrm{s} & \sim((\mathrm{w} \wedge \mathrm{q}) \vee \mathrm{s}) \
\hline \mathrm{T} & \mathrm{T} & \mathrm{T} & \mathrm{F} \
\mathrm{T} & \mathrm{T} & \mathrm{F} & \mathrm{F} \
\mathrm{T} & \mathrm{F} & \mathrm{T} & \mathrm{F} \
\mathrm{T} & \mathrm{F} & \mathrm{F} & \mathrm{T} \
\mathrm{F} & \mathrm{T} & \mathrm{T} & \mathrm{F} \
\mathrm{F} & \mathrm{T} & \mathrm{F} & \mathrm{T} \
\mathrm{F} & \mathrm{F} & \mathrm{T} & \mathrm{F} \
\mathrm{F} & \mathrm{F} & \mathrm{F} & \mathrm{T}
\end{array}$

Accepted Answer

The correct truth table for $\sim((w \wedge q) \vee s)$ reflects the negation of the disjunction between $w \wedge q$ and $s$. The output is false when either $w \wedge q$ is true or $s$ is true, and true otherwise, as shown in option B.

Question 18

Letting r stand for "The food is good," p stand for "I eat too much," and q stand for "I'll exercise," write the following in symbolic form.
-I'll exercise if I eat too much. 
A) $q \wedge p$
B) $p \rightarrow q$
C) $\mathrm { p \vee q }$
D) $q \rightarrow p$

Accepted Answer

The statement "I'll exercise if I eat too much" is a conditional statement where "I eat too much" is the condition (p) that leads to the consequence "I'll exercise" (q). This is represented symbolically as $p \rightarrow q$.

Question 19

Restate in a logically equivalent form.
-If it is clean, then it was washed.
A)If it is clean, then it was not washed.
B)If it is not clean, then it was not washed.
C)If it is clean, then it was washed.
D)If it was not washed, then it is not clean.

Accepted Answer

The contrapositive of a conditional statement is logically equivalent to the original statement. The contrapositive of "If it is clean, then it was washed" is "If it was not washed, then it is not clean."

Question 20

Letting r stand for "The food is good," p stand for "I eat too much," and q stand for "I'll exercise," write the following in symbolic form.
-The food is good and if I eat too much, then I'll exercise. 
A) $( r \vee p ) \rightarrow q$
B) $( r \wedge p ) \rightarrow q$
C) $r \wedge ( p \rightarrow q )$
D) $( r \rightarrow p ) \vee q$

Accepted Answer

The statement "The food is good and if I eat too much, then I'll exercise" can be broken down into two parts: "The food is good" (r) and "if I eat too much, then I'll exercise" (p → q). Combining these with 'and' gives $r \wedge ( p \rightarrow q )$.

Quiz 2: Introduction to Logic and Sets

Restate in a logically equivalent form. -The flowers are not blooming or it is not winter.

Construct a truth table for the statement. - $(t \wedge p) \vee(\sim t \wedge \sim p)$

Letting r stand for "The food is good," p stand for "I eat too much," and q stand for "I'll exercise," write the following in symbolic form. -If I eat too much, then I'll exercise.

Letting r stand for "The food is good," p stand for "I eat too much," and q stand for "I'll exercise," write the following in symbolic form. -I'll exercise if I don't eat too much.

Construct a truth table for the statement. - $\sim(\mathrm{s} \vee \mathrm{t}) \wedge \sim(\mathrm{t} \wedge \mathrm{s})$

Letting r stand for "The food is good," p stand for "I eat too much," and q stand for "I'll exercise," write the following in symbolic form. -If I exercise, then I won't eat too much.

Construct a truth table for the statement. - $\sim ( \sim ( \mathrm { s } \vee \mathrm { p } ) )$

Restate in a logically equivalent form. -It is not true that both this book is interesting and the book is about stars.

Restate in a logically equivalent form. -It is not true that today I both went to school and read a book.

Letting r stand for "The food is good," p stand for "I eat too much," and q stand for "I'll exercise," write the following in symbolic form. -If I exercise, then the food won't be good and I won't eat too much.

Letting r stand for "The food is good," p stand for "I eat too much," and q stand for "I'll exercise," write the following in symbolic form. -If the food is not good, I won't eat too much.

Letting r stand for "The food is good," p stand for "I eat too much," and q stand for "I'll exercise," write the following in symbolic form. -If the food is good or if I eat too much, I'll exercise.

Construct a truth table for the statement. - $(p \wedge w) \wedge(\sim w \vee t)$

Letting r stand for "The food is good," p stand for "I eat too much," and q stand for "I'll exercise," write the following in symbolic form. -If the food is good and if I eat too much, then I'll exercise.

Construct a truth table for the statement. - $w \vee ( w \wedge \sim w )$

Restate in a logically equivalent form. -If a number is divisible by 4 , then it is divisible by 2 .

Construct a truth table for the statement. - $\sim((w \wedge \mathrm{q}) \vee \mathrm{S})$

Letting r stand for "The food is good," p stand for "I eat too much," and q stand for "I'll exercise," write the following in symbolic form. -I'll exercise if I eat too much.

Restate in a logically equivalent form. -If it is clean, then it was washed.

Letting r stand for "The food is good," p stand for "I eat too much," and q stand for "I'll exercise," write the following in symbolic form. -The food is good and if I eat too much, then I'll exercise.