15.1: Chapter 2

Your Turn

2.1

1. A logical statement is a sentence that can be identified as being true or false. This sentence is a logical statement because it can be identified as false.
Logical statement, false.

2. A logical statement is a sentence that can be identified as being true or false. This sentence is a logical statement because it can be identified as true.
Logical statement, true.

3. A logical statement is a sentence that can be identified as being true or false. This sentence is a not a logical statement because questions cannot be identified as either true or false.
Not a logical statement, questions cannot be determined to be either true or false.

2.2

1. Choose a lowercase letter. The most common choices are \(p, q\), and \(r\). You are not being asked to determine the truth value.
p: The movie Gandhi won the Academy Award for Best Picture in 1982.
p: The movie Gandhi won the Academy Award for Best Picture in 1982.

2. Choose a lowercase letter. The most common choices are \(p, q\), and \(r\). You are not being asked to determine the truth value.
\(q\) : Soccer is the most popular sport in the world.
\(q\) : Soccer is the most popular sport in the world.

3. Choose a lowercase letter. The most common choices are \(p, q\), and \(r\). You are not being asked to determine the truth value.
\(r\) : All oranges are citrus fruits.
\(r\) : All oranges are citrus fruits.

2.3
1. The negation of a logical statement has the opposite truth value of the original statement.

Original statement: Ted Cruz was not born in Texas.
Remove "not."
Negation: Ted Cruz was born in Texas.
Ted Cruz was born in Texas.

2. The negation of a logical statement has the opposite truth value of the original statement.

Original statement: Adele has a lovely singing voice.
Add "not."
Negation: Adele does not have a lovely singing voice.
Adele does not have a beautiful voice.

3. The negation of a logical statement has the opposite truth value of the original statement.

Original statement: Leaves convert carbon dioxide to oxygen during the process of photosynthesis.
Add "not."
Negation: Leaves do not convert carbon dioxide to oxygen during the process of photosynthesis.
Leaves do not convert carbon dioxide to oxygen during the process of photosynthesis.

2.4

1. The negation of a logical statement has the opposite truth value of the original statement.

Original statement: \(\sim p\)
To negate a negative statement, remove the tilde, since the opposite of a negative is a positive.
Negation: \(\sim(\sim p)=p\)
p

2. The negation of a logical statement has the opposite truth value of the original statement.

Original statement: \(q\)
To negate an affirmative statement, add a tilde.
Negation: \(\sim q\)
\(\sim q\)

3. The negation of a logical statement has the opposite truth value of the original statement.

Original statement: \(r\)
To negate an affirmative statement, add a tilde.
Negation: \(\sim r\)
\(\sim r\)

2.5

1. "Wonder Woman is stronger than Captain Marvel" is the negation of "Wonder Woman is not stronger than Captain Marvel."

Thus, symbolically you write \(\sim(\sim p)=p\).
p
2. The statement \(r\) represents "Woody and Buzz Lightyear are best friends."

The negation \(\sim r\) is translated "Woody and Buzz Lightyear are not best friends."
Woody and Buzz Lightyear are not best friends.

2.6

1. It would be incorrect to say that the sum of all consecutive integers results in a prime number because 0 and \(1=1\), which is not prime. You can say that "The sum of some consecutive integers results in a prime number."
The sum of some consecutive integers results in a prime number.

2. Based on the premises provided, no birds give live birth to their young. This statement, however, is false, as some birds do give birth to live young.

No birds give live birth to their young.

3. All squares are rectangles. All rectangles are parallelograms, so a square is a parallelogram. All parallelograms have four sides, so a square has four sides. Thus, all squares have four sides.

You could draw a Venn diagram that shows this. Squares are a subset of rectangles. Rectangles are a subset of parallelograms. Parallelograms are a subset of things with four sides. Thus, squares are a subset of things with four sides.

All squares are parallelograms and have four sides.

2.7

1. You negate a "Some...not" statement by replacing "Some...not" with "All."

Original statement: Some apples are not sweet.
Negation: All apples are sweet.
All apples are sweet.

2. You negate a "No" statement by replacing "No" with "Some."

Original statement: No triangles are squares.
Negation: Some triangles are squares.
Some triangles are squares.

3. You negate a "some" statement by replacing "Some" with "No."

Original statement: Some vegetables are green.
Negation: No vegetables are green.
No vegetables are green.

2.8

1. Negation; \(\sim\).
2. Conjunction; \(\wedge\).
3. Biconditional; \(\leftrightarrow\).

2.9

1. Today we will go skiing, but last night it did not snow.

Both "and" and "but" use the conjunction connective.
The given statements:
\(p\) : Last night it snowed.
\(q\) : Today we will go skiing.
Break down the compound statement into its components.

 

Today we will go skiing,	but	not	last night it snowed
q	\(\wedge\)	\(\sim\)	p

2. \(q \leftrightarrow p\)

Today we will go skiing if and only if it snowed last night.
The given statements:
\(p\) : Last night it snowed.
\(q\) : Today we will go skiing.
Break down the compound statement into its components.
\begin{tabular}{|l|l|l|}
\hline Today we will go skiing & if and only if & it snowed last night \\
\hline \(\boldsymbol{q}\) & \(\leftrightarrow\) & \(p\) \\
\hline
\end{tabular}

\[
q \leftrightarrow p
\nonumber \]

3. \(p \vee \sim q\)

Last night it snowed or today we will not go skiing.
The given statements:
\(p\) : Last night it snowed.
\(q\) : Today we will go skiing.
Break down the compound statement into its components.

4. \(p \rightarrow q\)

If it snowed last night, then today we will go skiing.
The given statements:
\(p\) : Last night it snowed.
\(q\) : Today we will go skiing.

Last night it snowed	or	not	today we will go skiing,
p	\(\vee\)	\(\sim\)	q

\(p \vee \sim q\)

2.10

1. \(\sim r \rightarrow(p \vee q)\)

Given statements:
\(p\) : My roommates ordered pizza.
\(q\) : I ordered wings.
\(r\) : Our friends came over to watch the game.
Translate \(\sim r\) as :Our friends did not come over to watch the game.
\(\rightarrow\) is the conditional, which will be translated "if...then."
\(V\) is the inclusive or.
The comma before "then" shows that the statements in parentheses belong together.
If our friends did not come over to watch the game, then my roommates ordered pizza or I ordered wings. If our friends did not come over to watch the game, then my roommates ordered pizza or I ordered wings.

2. \((p \wedge q) \rightarrow r\)
\(\Lambda\) is the conjunction, which will be translated "and."
\(\rightarrow\) is the conditional, which will be translated "if...then."
The comma before "then" shows that the statements in parentheses belong together.
Given statements:
\(p\) : My roommates ordered pizza.
\(q\) : I ordered wings.
\(r\) : Our friends came over to watch the game.
If my roommates ordered pizza and I ordered wings, then our friends came over to watch the game.
If my roommates ordered pizza and I ordered wings, then our friends came over to watch the game.

3. \(\sim(p \vee r)\)
\(\sim\) is the negation symbol. Replace the symbol with "It is not the case that..."
\(V\) is the inclusive or.
Given statements:
\(p\) : My roommates ordered pizza.
\(q\) : I ordered wings.
\(r\) : Our friends came over to watch the game.
It is not the case that my roommates ordered pizza or our friends came over to watch the game.
It is not the case that my roommates ordered pizza or our friends came over to watch the game.

2.11

1. \(((p \vee q) \wedge(\sim r))\)

The order of logical operations is
1. Parentheses
2. Negation
3. Disjunction, conjunction
4. Conditional
5. Biconditional

For same order operations, work left to right.
The highest order operation in this expression is negation, so add parentheses around the negation.

\[
p \vee q \wedge(\sim r)
\nonumber \]

Disjunction and conjunction are the same order, so work left to right. First evaluate the exclusive or.

\[
(p \vee q) \wedge(\sim r)
\nonumber \]

Then finish by evaluating the conjunction.

\[
((p \vee q) \wedge(\sim r))
\nonumber \]

2. \(((\sim p) \rightarrow(q \vee r))\)

The order of logical operations is
1. Parentheses
2. Negation
3. Disjunction, conjunction
4. Conditional
5. Biconditional

For same order operations, work left to right.

\[
\sim p \rightarrow q \vee r
\nonumber \]

The highest order operation in this expression is negation, so add parentheses around the negation.

\[
(\sim p) \rightarrow q \vee r
\nonumber \]

The next highest order operation is the disjunction. Add parentheses around the disjunction.

\[
(\sim p) \rightarrow(q \vee r)
\nonumber \]

Finally, you can evaluate the conditional.

\[
((\sim p) \rightarrow(q \vee r))
\nonumber \]

3. \(((\sim p) \vee(\sim q)) \leftrightarrow(\sim(p \wedge q))\); this is another example of De Morgan's Laws and it is always true. The order of logical operations is

1. Parentheses
2. Negation
3. Disjunction, conjunction
4. Conditional
5. Biconditional

For same order operations, work left to right.

\(\sim p \vee \sim q \leftrightarrow \sim(p \wedge q)\)
The right side of the biconditional already has parentheses. The next step in the order of operation is negation, so add parentheses around the three negations.

\[
(\sim p) \vee(\sim q) \leftrightarrow(\sim(p \wedge q))
\nonumber \]

The next step in the order of operations is disjunction/conjunction. Add parentheses around the disjunction.

\[
((\sim p) \vee(\sim q)) \leftrightarrow(\sim(p \wedge q))
\nonumber \]

The lowest order operation is the biconditional. Finish by evaluating the biconditional.

\[
(((\sim p) \vee(\sim q)) \leftrightarrow(\sim(p \wedge q)))
\nonumber \]

2.12

1. The negation is \(p: 3 \times 5 \neq 14\)

Since \(3 \times 5=15\), the negation is true.
\(p: 3 \times 5 \neq 14\), true

2. The negation is \(q\) : No house are built with bricks.

The negation is false since some houses are built of bricks.
\(q\) : No houses are built with bricks; false

3. The negation is \(\sim r\) : Abuja is not the capital of Nigeria.

The negation is a false statement since Abuja is the capital of Nigeria.
\(\sim r\) : Abuja is not the capital of Nigeria; false

2.13

1. Given statements:
\(p\) : Yellow is a primary color.
\(p\) is true.
\(q\) : Blue is a primary color.
\(q\) is true.
The \(\wedge\) symbol represents the conjunction and is translated "and" or "but."
A conjunction is only true when both statements are true.
\(p \wedge q\)
True \(\wedge\) True \(=\) True
True

2. Given statements:
\(p\) : Yellow is a primary color.
\(p\) is true.
\(r\) : Green is a primary color.
\(r\) is false.
\(\sim r\) is true.
\(\sim r \wedge p\)
A conjunction is only true when both statements are true.
True \(\wedge\) True = True
False

3. Given statements:
\(q\) : Blue is a primary color.
\(q\) is true.
\(r\) : Green is a primary color.
\(r\) is false.
\(q \wedge r\)
A conjunction is only true when both statements are true.
True \(\wedge\) False \(=\) False
True

2.14

1. Given statements:
\(p\) : Yellow is a primary color.
\(p\) is true.
\(q\) : Blue is a primary color.
\(q\) is true.

\[
p \vee q
\nonumber \]

A disjunction is only false when both statements are false.
True \(\vee\) True \(=\) True
True

2. Given statements:
\(p\) : Yellow is a primary color.
\(p\) is true.
\(\sim p\) is false.
\(r\) : Green is a primary color.
\(r\) is false.
\(\sim p \vee r\)
A disjunction is only false when both statements are false.
False \(\vee\) False \(=\) False
False

3. Given statements:
\(q\) : Blue is a primary color.
\(q\) is true.
\(r\) : Green is a primary color.
\(r\) is false.
\(q \vee r\)
A disjunction is only false when both statements are false.
True \(\vee\) False \(=\) True
True

2.15

1. Given statements:
\(p\) : Yellow is a primary color.
\(p\) is true.
\(q\) : Blue is a primary color.
\(q\) is true.
\(r\) : Green is a primary color.
\(r\) is false.

\(p\)	\(q\)	\(r\)	\(\text{~}q\)	\(\text{~}q \wedge p\)	\((\text{~}q \wedge p) \vee r\)
T	T	F	F	F	F

false

2. Given statements:
\(p\) : Yellow is a primary color. \(p\) is true.
\(q\) : Blue is a primary color.
\(q\) is true.
\(r\) : Green is a primary color.
\(r\) is false.

\(p\)	\(q\)	\(r\)	\(\text{~}r\)	\(p \vee q\)	\((p \vee q) \wedge (\text{~}r)\)
T	T	F	T	T	T

true

3. Given statements:
\(p\) : Yellow is a primary color. \(p\) is true.
\(q\) : Blue is a primary color.
\(q\) is true.
\(r\) : Green is a primary color.
\(r\) is false.

\(p\)	\(q\)	\(r\)	\((p \wedge r)\)	\(\text{~}(p \wedge r)\)	\(\text{~}(p \wedge r) \wedge q\)
T	T	F	F	T	T

true

2.16

1.

\(p\)	\(q\)	\(\text~q\)	\(p \wedge \text{~}q\)
T	T	F	F
T	F	T	T
F	T	F	F
F	F	T	F

2.

\(p\)	\(q\)	\(p \vee q\)	\(\text{~}(p \vee q)\)
T	T	T	F
T	F	T	F
F	T	T	F
F	F	F	T

3.

\(p\)	\(q\)	\(r\)	\(\text{~}q\)	\(p\, \wedge \text{~}q\)	\((p \wedge \text{~}q) \vee r\)
T	T	T	F	F	T
T	T	F	F	F	F
T	F	T	T	T	T
T	F	F	T	T	T
F	T	T	F	F	T
F	T	F	F	F	F
F	F	T	T	F	T
F	F	F	T	F	F

2.17

1. Valid

\(p\)	\(\text{~}{p}\)	\({p} \vee \text{~}{p}\)
T	F	T
F	T	T

2. Not valid

\(p\)	\(q\)	\(\text{~}{p}\)	\(\text{~}{q}\)	\(\text{~}p \vee ~q\)
T	T	F	F	F
T	F	F	T	T
F	T	T	F	T
F	F	T	T	T

2.18

1. False

\(p\)	\(q\)	\({p} \to {q}\)
T	F	F

2. True

\(p\)	\(q\)	\(\text{~}{q}\)	\({p} \to \text{~}{q}\)
T	F	T	T

3. True

\(p\)	\(q\)	\(\text{~}{p}\)	\(\text{~}{p} \to \text{~}q\)
T	F	F	T

2.19

1. Valid

\(p\)	\(q\)	\(\text{~}{p}\)	\(\text{~}{p} \vee {q}\)	\({q} \to \left( {\text{~}{p} \vee {q}} \right)\)
T	T	F	T	T
T	F	F	F	T
F	T	T	T	T
F	F	T	T	T

2. Not valid

\(p\)	\(q\)	\(\text{~}{p}\)	\({q} \wedge {p}\)	\(\sim p \rightarrow(q \wedge p)\)
T	T	F	T	T
T	F	F	F	T
F	T	T	F	F
F	F	T	F	F

2.20

1. False

\(p\)	\(q\)	\({p} \leftrightarrow {q}\)
T	F	F

2. True

\(p\)	\(q\)	\(\text{~}{q}\)	\({p} \leftrightarrow \text{~}{q}\)
T	F	T	T

3. True

\(p\)	\(q\)	\(\text{~}{p}\)	\(\text{~}{p} \leftrightarrow {q}\)
T	F	F	T

2.21

1. Valid

\(p\)	\(q\)	\({p} \wedge {q}\)	\(\sim(p \wedge q)\)	\(\text{~}{p}\)	\(\text{~}{q}\)	\(\sim p \vee \sim q\)	\(\sim(p \wedge q) \leftrightarrow(\sim p \vee \sim q)\)
T	T	T	F	F	F	F	T
T	F	F	T	F	T	T	T
F	T	F	T	T	F	T	T
F	F	F	T	T	T	T	T

2. Not Valid

\(p\)	\(q\)	\(\text{~}{p}\)	\({q} \wedge {p}\)	\({ }^{\sim} p \leftrightarrow(q \wedge p)\)
T	T	F	T	F
T	F	F	F	T
F	T	T	F	F
F	F	T	F	F

3. Valid

\(p\)	\(q\)	\({p} \to {q}\)	\(\text{~}{p}\)	\(\text{~}{p} \vee {q}\)	\((p \rightarrow q) \leftrightarrow(\sim p \vee q)\)
T	T	T	F	T	T
T	F	F	F	F	T
F	T	T	T	T	T
F	F	T	T	T	T

4. Valid

\(p\)	\(q\)	\(r\)	\(\text{~}{p}\)	\(\text{~}{q}\)	\({p} \wedge {q}\)	\(\left( ParseError: EOF expected (click for details) Callstack: at (Bookshelves/Applied_Mathematics/Contemporary_Mathematics_(OpenStax)/15:_Answer_Key/15.01:_Chapter_2), /content/body/div[19]/table/thead/tr/th[7]/span, line 1, column 2 \right) \to {r}\)	\(\text{~}{p} \vee \text{~}{q}\)	\((\sim p \vee \sim q) \vee r\)	\((p \wedge q \rightarrow r) \leftrightarrow(\sim p \vee \sim q \vee r)\)
T	T	T	F	F	T	T	F	T	T
T	T	F	F	F	T	F	F	F	T
T	F	T	F	T	F	T	T	T	T
T	F	F	F	T	F	T	T	T	T
F	T	T	T	F	F	T	T	T	T
F	T	F	T	F	F	T	T	T	T
F	F	T	T	T	F	T	T	T	T
F	F	F	T	T	F	T	T	T	T

2.22

1. \(p \to q\) is logically equivalent to \(\text{~}q \to\text{~}p.\)

\({p}\)	\({q}\)	\({p} \to {q}\)	\(\text{~}{q}\)	\(\text{~}{p}\)	\(\text{~}{q} \to \text{~}{p}\)	\((p \rightarrow q) \leftrightarrow(\sim q \rightarrow \sim p)\)
T	T	T	F	F	T	T
T	F	F	T	F	F	T
F	T	T	F	T	T	T
F	F	T	T	T	T	T

2. \(p \to q\) is not logically equivalent to \(p \vee \text{~}q.\)

\(p\)	\(q\)	\({p} \to {q}\)	\(\text{~}{q}\)	\({p}\rm \vee \text{~}{q}\)	\((p \rightarrow q) \leftrightarrow(p \vee \sim q)\)
T	T	T	F	T	T
T	F	F	T	T	F
F	T	T	F	F	F
F	F	T	T	T	T

2.23

1.

If Elvis Presley wore capes, then some superheroes wear capes.

2.

If some superheroes wear capes, then Elvis Presley wore capes.

3.

If Elvis Presley did not wear capes, then no superheroes wear capes.

4.

If no superheroes wear capes, then Elvis Presley did not wear capes.

2.24

1. \(p\): Dora is an explorer.

2. \(q\): Boots is a monkey.

3. Inverse

4. Converse

5. Converse

2.25

1. If my friend does not live in California, then my friend lives in San Francisco. True.

2. If my friend does not live in San Francisco, then my friend lives in California. True.

3. If my friend lives in California, then my friend does not live in San Francisco. False.

2.26

1. Jackie did not play softball and she did not run track.

2.

Brandon did not study for his certification exam, or he did not pass his exam.

2.27

1. Edna Mode made a new superhero costume, and it includes a cape.

2.

I had pancakes for breakfast, and I did not use maple syrup.

2.28

1. Some people like ice cream, but ice cream makers will not make a profit.

2. Raquel cannot play video games, but somebody will play video games.

2.29

1. Eric needs to replace the light bulb, and Marcos did not leave the light bulb on all night, and Dan did not break the light bulb.

2. Trenton went to school, and Regina went to work, and Merika did not clean the house.

2.30

1.

\(p\)	\(q\)	\({p} \vee {q}\)	\(\text{~}\left( ParseError: EOF expected (click for details) Callstack: at (Bookshelves/Applied_Mathematics/Contemporary_Mathematics_(OpenStax)/15:_Answer_Key/15.01:_Chapter_2), /content/body/div[29]/div/table/thead/tr/th[4]/span, line 1, column 2 \right)\)	\(\text{~}p\)	\(\text{~}q\)	\(\text{~}p \wedge \text{~}q\)	\(\text{~}({p} \vee {q}) \leftrightarrow \left( {\text{~}p \wedge \text{~}q} \right)\)
T	T	T	F	F	F	F	T
T	F	T	F	F	T	F	T
F	T	T	F	T	F	F	T
F	F	F	T	T	T	T	T

2.31

1. Some people like history.

2.

Some people do not like reading.

3.

The polygon is not an octagon.

2.32

1. My classmate does not like history.

2. Homer likes to read.

3. The polygon does not have five sides.

2.33

1. If my roommate does not go to work, then they will not be able to pay their bills.

2. If penguins cannot fly, then we will watch the news.

3. If Marcy goes to the movies, then she will buy water.

Check Your Understanding

1. logical statement

2.

negation

3. \(\text{~}p\)

4.

\(p\)

5.

premises

6.

Inductive

7.

quantifiers

8. Some giraffes are not tall.

9. compound statement

10.

connective

11.

biconditional, \(\leftrightarrow\)

12.

Parentheses, \((\,)\)

13.

Conjunction, \(\wedge\); disjunction, \(\vee\) (in any order)

14.

valid

15.

true

16.

truth table

17.

four

18.

two

19. one-way contract
20. conclusion
21. If the hypothesis of a conditional statement is false, the conditional statement is true. A false hypothesis can have either a true or a false conclusion. In both situations, the conditional is true. hypothesis
22. biconditional
23. biconditional
24. true
25. always true, valid, or a tautology.
26. conditional
27. logically equivalent
28. inverse
29. converse, inverse
30. \(\sim p \vee \sim q\)
31. \(\sim p \wedge \sim q\)
32. \(p \wedge \sim q\)
33. De Morgan's Laws
34. premise
35. A logical argument is valid if its conclusion follows from the premises. The validity is not affected by whether the premises are true or not. valid
36. inductive
37. deductive
38. fallacy
39. sound