Question 1

Write the compound statement in symbols.
Let r = ʺThe food is good,ʺ p = ʺI eat too much,ʺ q = ʺIʹll exercise.ʺ
-If the food is good, then I eat too much.

Accepted Answer

A)  $r \wedge p$ 
B)  $p \rightarrow r$ 
C)  $\mathrm { r } \rightarrow \mathrm { p }$ 
D)  $\mathrm { p } \rightarrow \mathrm { q }$ 
A)  $r \wedge p$ 
B)  $p \rightarrow r$ 
C)  $\mathrm { r } \rightarrow \mathrm { p }$ 
D)  $\mathrm { p } \rightarrow \mathrm { q }$

Question 2

Write the contrapositive of the statement. Then use the contrapositive to determine whether to conditional statement is true or false.
-If $\frac { 1 } { \mathrm { n } }$ is not an integer, then $\mathrm { n }$ is not an integer.

Accepted Answer

A)  If $\mathrm { n }$ is an integer, then $\frac { 1 } { \mathrm { n } }$ is an integer. false 
B)  If $\frac { 1 } { n }$ is an integer, then $n$ is an integer. false 
C)  If $\frac { 1 } { n }$ is an integer, then $n$ is an integer. true 
D)  If $\mathrm { n }$ is an integer, then $\frac { 1 } { n }$ is an integer. true 
A)  If $\mathrm { n }$ is an integer, then $\frac { 1 } { \mathrm { n } }$ is an integer. false 
B)  If $\frac { 1 } { n }$ is an integer, then $n$ is an integer. false 
C)  If $\frac { 1 } { n }$ is an integer, then $n$ is an integer. true 
D)  If $\mathrm { n }$ is an integer, then $\frac { 1 } { n }$ is an integer. true

Question 3

Use truth tables to test the validity of the argument.
-$$\begin{array} { l } 
\mathrm { p } \rightarrow \mathrm { q } \
\frac { \sim \mathrm { p } } { \therefore \sim \mathrm { q } }
\end{array}$$

Accepted Answer

A)  Valid 
B)  Invalid 
A)  Valid 
B)  Invalid

Question 4

Write the compound statement in symbols.
Let r = ʺThe food is good,ʺ p = ʺI eat too much,ʺ q = ʺIʹll exercise.ʺ
-If I eat too much, then Iʹll exercise.

Accepted Answer

A)  $q \rightarrow p$ 
B)  $r \rightarrow p$ 
C)  $\mathrm { p } \vee \mathrm { q }$ 
D)  $\mathrm { p } \rightarrow \mathrm { q }$ p 
A)  $q \rightarrow p$ 
B)  $r \rightarrow p$ 
C)  $\mathrm { p } \vee \mathrm { q }$ 
D)  $\mathrm { p } \rightarrow \mathrm { q }$ p

Question 5

Write a negation of the statement.
-Everyone is asleep.

Accepted Answer

A)  Everyone is not asleep. 
B)  None are asleep. 
C)  Everyone is awake. 
D)  Not everyone is asleep. 
A)  Everyone is not asleep. 
B)  None are asleep. 
C)  Everyone is awake. 
D)  Not everyone is asleep.

Question 6

Use truth tables to test the validity of the argument.
-$\begin{array} { l } 
\mathrm { p } \sim \mathrm { q } \
 \sim \mathrm { q }\
\hline\therefore \sim \mathrm { p } 
\end{array}$

Accepted Answer

A)  Valid 
B)  Invalid 
A)  Valid 
B)  Invalid

Question 7

Write the indicated statement. Use De Morgan's Laws if necessary.
-If the moon is out, then we will start a campfire and we will roast marshmallows.
Inverse

Accepted Answer

A)  If we start a campfire and we roast marshmallows, then the moon is out. 
B)  If the moon is not out, then we will not start a campfire or we will not roast marshmallows. 
C)  If we do not start a campfire or we do not roast marshmallows, then the moon is not out. 
D)  If the moon is not out, then we will start a campfire but we will not roast marshmallows. 
A)  If we start a campfire and we roast marshmallows, then the moon is out. 
B)  If the moon is not out, then we will not start a campfire or we will not roast marshmallows. 
C)  If we do not start a campfire or we do not roast marshmallows, then the moon is not out. 
D)  If the moon is not out, then we will start a campfire but we will not roast marshmallows.

Question 8

Use truth tables to test the validity of the argument.
-$\begin{array} { l } 
\sim \mathrm { p } \rightarrow \mathrm { q } \
\sim \mathrm { q } \rightarrow \mathrm { p }\
\hline  \therefore \mathrm { p } \vee \mathrm { q }  \
\end{array}$

Accepted Answer

A)  Valid 
B)  Invalid 
A)  Valid 
B)  Invalid

Question 9

Write the compound statement in symbols. Then construct a truth table for the symbolic statement.
Let r = ʺThe food is good,ʺ p = ʺI eat too much,ʺ q = ʺIʹll exercise.ʺ
-If the food is good, then I eat too much.

Accepted Answer

A)  $r \rightarrow p$
$\begin{array} { c c c } 
\mathrm { r } & \mathrm { p } & \mathrm { r } \rightarrow \mathrm { p } \
\hline \mathrm { T } & \mathrm { T } & \mathrm { F } \
\mathrm { T } & \mathrm { F } & \mathrm { T } \
\mathrm { F } & \mathrm { T } & \mathrm { T } \
\mathrm { F } & \mathrm { F } & \mathrm { F }
\end{array}$ 
B) $r \rightarrow p$
$\begin{array} { c c c } 
\mathrm { r } & \mathrm { p } & \mathrm { r } \rightarrow \mathrm { p } \
\hline \mathrm { T } & \mathrm { T } & \mathrm { T } \
\mathrm { T } & \mathrm { F } & \mathrm { F } \
\mathrm { F } & \mathrm { T } & \mathrm { T } \
\mathrm { F } & \mathrm { F } & \mathrm { T }
\end{array}$ 
C) $P \rightarrow r$
$\begin{array} { c c c } 
\mathrm { r } & \mathrm { p } & \mathrm { p } \rightarrow \mathrm { r } \
\hline \mathrm { T } & \mathrm { T } & \mathrm { T } \
\mathrm { T } & \mathrm { F } & \mathrm { F } \
\mathrm { F } & \mathrm { T } & \mathrm { T } \
\mathrm { F } & \mathrm { F } & \mathrm { T }
\end{array}$ 
D) $P \rightarrow \mathbf{r}$
$\begin{array} { c c r } 
\mathrm { r } & \mathrm { p } & \mathrm { p } \rightarrow \mathrm { r } \
\hline \mathrm { T } & \mathrm { T } & \mathrm { T } \
\mathrm { T } & \mathrm { F } & \mathrm { T } \
\mathrm { F } & \mathrm { T } & \mathrm { F } \
\mathrm { F } & \mathrm { F } & \mathrm { T }
\end{array}$ 
A)  $r \rightarrow p$
$\begin{array} { c c c } 
\mathrm { r } & \mathrm { p } & \mathrm { r } \rightarrow \mathrm { p } \
\hline \mathrm { T } & \mathrm { T } & \mathrm { F } \
\mathrm { T } & \mathrm { F } & \mathrm { T } \
\mathrm { F } & \mathrm { T } & \mathrm { T } \
\mathrm { F } & \mathrm { F } & \mathrm { F }
\end{array}$ 
B) $r \rightarrow p$
$\begin{array} { c c c } 
\mathrm { r } & \mathrm { p } & \mathrm { r } \rightarrow \mathrm { p } \
\hline \mathrm { T } & \mathrm { T } & \mathrm { T } \
\mathrm { T } & \mathrm { F } & \mathrm { F } \
\mathrm { F } & \mathrm { T } & \mathrm { T } \
\mathrm { F } & \mathrm { F } & \mathrm { T }
\end{array}$ 
C) $P \rightarrow r$
$\begin{array} { c c c } 
\mathrm { r } & \mathrm { p } & \mathrm { p } \rightarrow \mathrm { r } \
\hline \mathrm { T } & \mathrm { T } & \mathrm { T } \
\mathrm { T } & \mathrm { F } & \mathrm { F } \
\mathrm { F } & \mathrm { T } & \mathrm { T } \
\mathrm { F } & \mathrm { F } & \mathrm { T }
\end{array}$ 
D) $P \rightarrow \mathbf{r}$
$\begin{array} { c c r } 
\mathrm { r } & \mathrm { p } & \mathrm { p } \rightarrow \mathrm { r } \
\hline \mathrm { T } & \mathrm { T } & \mathrm { T } \
\mathrm { T } & \mathrm { F } & \mathrm { T } \
\mathrm { F } & \mathrm { T } & \mathrm { F } \
\mathrm { F } & \mathrm { F } & \mathrm { T }
\end{array}$

Question 10

Select letters to represent the simple statements and write each statement symbolically by using parentheses then indicate whether the statement is a negation, conjunction, disjunction, conditional, or biconditional.
-The lights are on if and only if it is not midnight or it is wintertime.

Accepted Answer

A)  $\mathrm { p } \leftrightarrow ( \sim \mathrm { q } \vee \mathrm { r } )$; biconditional 
B)  $( \mathrm { p } \leftrightarrow \sim \mathrm { q } ) \wedge \mathrm { r }$; conjunction 
C)  $( \mathrm { p } \leftrightarrow \sim \mathrm { q } ) \vee \mathrm { r }$; disjunction 
D)  $\mathrm { p } \leftrightarrow ( \sim \mathrm { q } \vee \mathrm { r } )$; conditional 
A)  $\mathrm { p } \leftrightarrow ( \sim \mathrm { q } \vee \mathrm { r } )$; biconditional 
B)  $( \mathrm { p } \leftrightarrow \sim \mathrm { q } ) \wedge \mathrm { r }$; conjunction 
C)  $( \mathrm { p } \leftrightarrow \sim \mathrm { q } ) \vee \mathrm { r }$; disjunction 
D)  $\mathrm { p } \leftrightarrow ( \sim \mathrm { q } \vee \mathrm { r } )$; conditional

Question 11

Let p represent a true statement, while q and r represent false statements. Find the truth value of the compound statement.
-$$p \wedge ( q \vee p )$$

Accepted Answer

A) True 
 B)False

Question 12

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

Accepted Answer

A) 
$\begin{array} { c c c } 
\mathrm { p } & \mathrm { q } & \sim ( \mathrm { p } \rightarrow \mathrm { q } ) \rightarrow ( \mathrm { p } \wedge \sim \mathrm { q } ) \
\hline \mathrm { T } & \mathrm { T } & \mathrm { T } \
\mathrm { T } & \mathrm { F } & \mathrm { F } \
\mathrm { F } & \mathrm { T } & \mathrm { T } \
\mathrm { F } & \mathrm { F } & \mathrm { T }
\end{array}$ 
B) 
$\begin{array} { c c c } 
\mathrm { p } & \mathrm { q } & \sim ( \mathrm { p } \rightarrow \mathrm { q } ) \rightarrow ( \mathrm { p } \wedge \sim \mathrm { q } ) \
\hline \mathrm { T } & \mathrm { T } & \mathrm { T } \
\mathrm { T } & \mathrm { F } & \mathrm { F } \
\mathrm { F } & \mathrm { T } & \mathrm { F } \
\mathrm { F } & \mathrm { F } & \mathrm { T }
\end{array}$ 
C) 
$\begin{array} { c c c } 
\mathrm { p } & \mathrm { q } & \sim ( \mathrm { p } \rightarrow \mathrm { q } ) \rightarrow ( \mathrm { p } \wedge \sim \mathrm { q } ) \
\hline \mathrm { T } & \mathrm { T } & \mathrm { T } \
\mathrm { T } & \mathrm { F } & \mathrm { F } \
\mathrm { F } & \mathrm { T } & \mathrm { F } \
\mathrm { F } & \mathrm { F } & \mathrm { F }
\end{array}$ 
D) 
$\begin{array} { c c c } 
\mathrm { p } & \mathrm { q } & \sim ( \mathrm { p } \rightarrow \mathrm { q } ) \rightarrow ( \mathrm { p } \wedge \sim \mathrm { q } ) \
\hline \mathrm { T } & \mathrm { T } & \mathrm { T } \
\mathrm { T } & \mathrm { F } & \mathrm { T } \
\mathrm { F } & \mathrm { T } & \mathrm { T } \
\mathrm { F } & \mathrm { F } & \mathrm { T }
\end{array}$ 
A) 
$\begin{array} { c c c } 
\mathrm { p } & \mathrm { q } & \sim ( \mathrm { p } \rightarrow \mathrm { q } ) \rightarrow ( \mathrm { p } \wedge \sim \mathrm { q } ) \
\hline \mathrm { T } & \mathrm { T } & \mathrm { T } \
\mathrm { T } & \mathrm { F } & \mathrm { F } \
\mathrm { F } & \mathrm { T } & \mathrm { T } \
\mathrm { F } & \mathrm { F } & \mathrm { T }
\end{array}$ 
B) 
$\begin{array} { c c c } 
\mathrm { p } & \mathrm { q } & \sim ( \mathrm { p } \rightarrow \mathrm { q } ) \rightarrow ( \mathrm { p } \wedge \sim \mathrm { q } ) \
\hline \mathrm { T } & \mathrm { T } & \mathrm { T } \
\mathrm { T } & \mathrm { F } & \mathrm { F } \
\mathrm { F } & \mathrm { T } & \mathrm { F } \
\mathrm { F } & \mathrm { F } & \mathrm { T }
\end{array}$ 
C) 
$\begin{array} { c c c } 
\mathrm { p } & \mathrm { q } & \sim ( \mathrm { p } \rightarrow \mathrm { q } ) \rightarrow ( \mathrm { p } \wedge \sim \mathrm { q } ) \
\hline \mathrm { T } & \mathrm { T } & \mathrm { T } \
\mathrm { T } & \mathrm { F } & \mathrm { F } \
\mathrm { F } & \mathrm { T } & \mathrm { F } \
\mathrm { F } & \mathrm { F } & \mathrm { F }
\end{array}$ 
D) 
$\begin{array} { c c c } 
\mathrm { p } & \mathrm { q } & \sim ( \mathrm { p } \rightarrow \mathrm { q } ) \rightarrow ( \mathrm { p } \wedge \sim \mathrm { q } ) \
\hline \mathrm { T } & \mathrm { T } & \mathrm { T } \
\mathrm { T } & \mathrm { F } & \mathrm { T } \
\mathrm { F } & \mathrm { T } & \mathrm { T } \
\mathrm { F } & \mathrm { F } & \mathrm { T }
\end{array}$

Question 13

Use truth tables to test the validity of the argument.
-$\begin{array}{l}
\mathrm{p} \rightarrow \mathrm{q} \
\mathrm{q}\
\hline\therefore \mathrm{p}
\end{array}$

Accepted Answer

A)  Invalid 
B)  Valid 
A)  Invalid 
B)  Valid

Question 14

Write the compound statement in words.
-Let $\mathbf { r } =$ "The puppy is trained."
$\mathrm { p } =$ "The puppy behaves well."
$q =$ "His owners are happy."
$r \wedge ( p - q )$

Accepted Answer

A)  The puppy is trained if the puppy behaves well and his owners are happy. 
B)  If the puppy is trained then the puppy behaves well, and his owners are happy. 
C)  The puppy is trained, and if the puppy behaves well then his owners are happy. 
D)  The puppy is trained, or if the puppy behaves well then his owners are happy. 
A)  The puppy is trained if the puppy behaves well and his owners are happy. 
B)  If the puppy is trained then the puppy behaves well, and his owners are happy. 
C)  The puppy is trained, and if the puppy behaves well then his owners are happy. 
D)  The puppy is trained, or if the puppy behaves well then his owners are happy.

Question 15

Write a negation of the statement.
-Some photographs are not displayed at this exhibition.

Accepted Answer

A)  No photographs are displayed at this exhibition. 
B)  Some photographs are displayed at this exhibition. 
C)  All photographs are displayed at this exhibition. 
D)  All photographs are not displayed at this exhibition. 
A)  No photographs are displayed at this exhibition. 
B)  Some photographs are displayed at this exhibition. 
C)  All photographs are displayed at this exhibition. 
D)  All photographs are not displayed at this exhibition.

Question 16

Determine the truth value for each simple statement. Then, using the truth values, give the truth value of the compound statement.
-$$3 + 1 = 11 \text { if and only if } 9 = 15 \text {. }$$

Accepted Answer

A) True 
 B)False

Question 17

Write an equivalent sentence for the statement.
-It is not true that you are a day late and a dollar short. (Hint: Use De Morganʹs laws.)

Accepted Answer

A)  You are not a day late or not a dollar short. 
B)  You are not a day late and a dollar short. 
C)  You are not a day late and not a dollar short. 
D)  You are a day late or not a dollar short. 
A)  You are not a day late or not a dollar short. 
B)  You are not a day late and a dollar short. 
C)  You are not a day late and not a dollar short. 
D)  You are a day late or not a dollar short.

Question 18

Use truth tables to test the validity of the argument.
-$\begin{array} { l } 
\mathrm { p } \vee \mathrm { q } \
 \mathrm { q }  \
\hline\mathrm { p } 
\end{array}$

Accepted Answer

A)  Invalid 
B)  Valid 
A)  Invalid 
B)  Valid

Question 19

Use DeMorganʹs laws or a truth table to determine whether the two statements are equivalent.
-$$\sim ( p \vee q ) , \sim p \wedge \sim q$$

Accepted Answer

A)  Equivalent 
B)  Not equivalent 
A)  Equivalent 
B)  Not equivalent

Question 20

Given p is true, q is true, and r is false, find the truth value of the statement.
-$$\sim [ ( \sim \mathrm { q } - \mathrm { r } )  \leftrightarrow ( \mathrm { q } \vee \mathrm { r } ) ]$$

Accepted Answer

A) True 
 B)False

Write the compound statement in symbols. Let r = ʺThe food is good,ʺ p = ʺI eat too much,ʺ q = ʺIʹll exercise.ʺ -If the food is good, then I eat too much.

Write the contrapositive of the statement. Then use the contrapositive to determine whether to conditional statement is true or false. -If $\frac { 1 } { \mathrm { n } }$ is not an integer, then $\mathrm { n }$ is not an integer.

Use truth tables to test the validity of the argument. - \rightarrow

Write the compound statement in symbols. Let r = ʺThe food is good,ʺ p = ʺI eat too much,ʺ q = ʺIʹll exercise.ʺ -If I eat too much, then Iʹll exercise.

Write a negation of the statement. -Everyone is asleep.

Use truth tables to test the validity of the argument. - \sim \sim \therefore\sim

Write the indicated statement. Use De Morgan's Laws if necessary. -If the moon is out, then we will start a campfire and we will roast marshmallows. Inverse

Use truth tables to test the validity of the argument. - \sim\rightarrow \sim\rightarrow \therefore\vee

Write the compound statement in symbols. Then construct a truth table for the symbolic statement. Let r = ʺThe food is good,ʺ p = ʺI eat too much,ʺ q = ʺIʹll exercise.ʺ -If the food is good, then I eat too much.

Select letters to represent the simple statements and write each statement symbolically by using parentheses then indicate whether the statement is a negation, conjunction, disjunction, conditional, or biconditional. -The lights are on if and only if it is not midnight or it is wintertime.

Let p represent a true statement, while q and r represent false statements. Find the truth value of the compound statement. - $p \wedge ( q \vee p )$

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

Use truth tables to test the validity of the argument. - \rightarrow \therefore

Write the compound statement in words. -Let $\mathbf { r } =$ "The puppy is trained." $\mathrm { p } =$ "The puppy behaves well." $q =$ "His owners are happy." $r \wedge ( p - q )$

Write a negation of the statement. -Some photographs are not displayed at this exhibition.

Determine the truth value for each simple statement. Then, using the truth values, give the truth value of the compound statement. - $3 + 1 = 11 \text { if and only if } 9 = 15 \text {. }$

Write an equivalent sentence for the statement. -It is not true that you are a day late and a dollar short. (Hint: Use De Morganʹs laws.)

Use truth tables to test the validity of the argument. - \vee

Use DeMorganʹs laws or a truth table to determine whether the two statements are equivalent. - $\sim ( p \vee q ) , \sim p \wedge \sim q$

Given p is true, q is true, and r is false, find the truth value of the statement. - $\sim [ ( \sim \mathrm { q } - \mathrm { r } ) \leftrightarrow ( \mathrm { q } \vee \mathrm { r } ) ]$

Critical Thinking Skills

Sets

Systems of Numeration

Number Theory and the Real Number System

Algebra, Graphs, and Functions

The Metric System

Geometry

Consumer Mathematics

Probability

Voting and Apportionment

Filters

Exam 3: Logic

Write the compound statement in symbols. Let r = ʺThe food is good,ʺ p = ʺI eat too much,ʺ q = ʺIʹll exercise.ʺ -If the food is good, then I eat too much.

Write the contrapositive of the statement. Then use the contrapositive to determine whether to conditional statement is true or false. -If 1n\frac { 1 } { \mathrm { n } }n1​ is not an integer, then n\mathrm { n }n is not an integer.

Use truth tables to test the validity of the argument. - \rightarrow

Write the compound statement in symbols. Let r = ʺThe food is good,ʺ p = ʺI eat too much,ʺ q = ʺIʹll exercise.ʺ -If I eat too much, then Iʹll exercise.

Write a negation of the statement. -Everyone is asleep.

Use truth tables to test the validity of the argument. - \sim \sim \therefore\sim

Write the indicated statement. Use De Morgan's Laws if necessary. -If the moon is out, then we will start a campfire and we will roast marshmallows. Inverse

Use truth tables to test the validity of the argument. - \sim\rightarrow \sim\rightarrow \therefore\vee

Write the compound statement in symbols. Then construct a truth table for the symbolic statement. Let r = ʺThe food is good,ʺ p = ʺI eat too much,ʺ q = ʺIʹll exercise.ʺ -If the food is good, then I eat too much.

Select letters to represent the simple statements and write each statement symbolically by using parentheses then indicate whether the statement is a negation, conjunction, disjunction, conditional, or biconditional. -The lights are on if and only if it is not midnight or it is wintertime.

Let p represent a true statement, while q and r represent false statements. Find the truth value of the compound statement. - p∧(q∨p)p \wedge ( q \vee p )p∧(q∨p)

Construct a truth table for the statement. - ∼(p→q)→(p∧∼q)\sim(p \rightarrow q) \rightarrow(p \wedge \sim q)∼(p→q)→(p∧∼q)

Use truth tables to test the validity of the argument. - \rightarrow \therefore

Write the compound statement in words. -Let r=\mathbf { r } =r= "The puppy is trained." p=\mathrm { p } =p= "The puppy behaves well." q=q =q= "His owners are happy." r∧(p−q)r \wedge ( p - q )r∧(p−q)

Write a negation of the statement. -Some photographs are not displayed at this exhibition.

Determine the truth value for each simple statement. Then, using the truth values, give the truth value of the compound statement. - 3+1=11 if and only if 9=15. 3 + 1 = 11 \text { if and only if } 9 = 15 \text {. }3+1=11 if and only if 9=15.

Write an equivalent sentence for the statement. -It is not true that you are a day late and a dollar short. (Hint: Use De Morganʹs laws.)

Use truth tables to test the validity of the argument. - \vee

Use DeMorganʹs laws or a truth table to determine whether the two statements are equivalent. - ∼(p∨q),∼p∧∼q\sim ( p \vee q ) , \sim p \wedge \sim q∼(p∨q),∼p∧∼q

Given p is true, q is true, and r is false, find the truth value of the statement. - ∼[(∼q−r)↔(q∨r)]\sim [ ( \sim \mathrm { q } - \mathrm { r } ) \leftrightarrow ( \mathrm { q } \vee \mathrm { r } ) ]∼[(∼q−r)↔(q∨r)]

Critical Thinking Skills

Sets

Systems of Numeration

Number Theory and the Real Number System

Algebra, Graphs, and Functions

The Metric System

Geometry

Consumer Mathematics

Probability

Voting and Apportionment

Filters

Write the contrapositive of the statement. Then use the contrapositive to determine whether to conditional statement is true or false. -If $\frac { 1 } { \mathrm { n } }$ is not an integer, then $\mathrm { n }$ is not an integer.

Let p represent a true statement, while q and r represent false statements. Find the truth value of the compound statement. - $p \wedge ( q \vee p )$

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

Write the compound statement in words. -Let $\mathbf { r } =$ "The puppy is trained." $\mathrm { p } =$ "The puppy behaves well." $q =$ "His owners are happy." $r \wedge ( p - q )$

Determine the truth value for each simple statement. Then, using the truth values, give the truth value of the compound statement. - $3 + 1 = 11 \text { if and only if } 9 = 15 \text {. }$

Use DeMorganʹs laws or a truth table to determine whether the two statements are equivalent. - $\sim ( p \vee q ) , \sim p \wedge \sim q$

Given p is true, q is true, and r is false, find the truth value of the statement. - $\sim [ ( \sim \mathrm { q } - \mathrm { r } ) \leftrightarrow ( \mathrm { q } \vee \mathrm { r } ) ]$