Contemporary Mathematics

# Chapter 2

2.1
1.
Logical statement, false.
2.
Logical statement, true.
3.
Not a logical statement, questions cannot be determined to be either true or false.
2.2
1.
$p{\text{:}}$ The movie Gandhi won the Academy Award for Best Picture in 1982.
2.
$q{\text{:}}$ Soccer is the most popular sport in the world.
3.
$r{\text{:}}$ All oranges are citrus fruits.
2.3
1.
Ted Cruz was born in Texas.
2.
Adele does not have a beautiful voice.
3.
Leaves do not convert carbon dioxide to oxygen during the process of photosynthesis.
2.4
1.
$p$
2.
$\text~q$
3.
$\text~r$
2.5
1.
$p$
2.
Woody and Buzz Lightyear are not best friends.
2.6
1.
The sum of some consecutive integers results in a prime number.
2.

No birds give live birth to their young.

3.

All squares are parallelograms and have four sides.

2.7
1.
All apples are sweet.
2.
Some triangles are squares.
3.
No vegetables are green.
2.8
1.
Negation; $\sim$.
2.
Conjunction; $\wedge$.
3.
Biconditional; $\leftrightarrow$.
2.9
1.
$q \wedge \text{~}p$
2.
$q \leftrightarrow p$
3.
$p \vee \text{~}q$
4.
$p \to q$
2.10
1.
If our friends did not come over to watch the game, then my roommates ordered pizza or I ordered wings.
2.
If my roommates ordered pizza and I ordered wings, then our friends came over to watch the game.
3.
It is not the case that my roommates ordered pizza or our friends came over to watch the game.
2.11
1.
$((p \vee q) \wedge (\text{~}r))$
2.
$((\text{~}p) \to (q \vee r))$
3.
$((\text{~}p) \vee (\text{~}q)) \leftrightarrow (\text{~}(p \wedge q))$; this is another example of De Morgan’s Laws and it is always true.
2.12
1.
$p{\text{:}}\,3 \times 5 \ne 14$, true
2.
$q$: No houses are built with bricks; false
3.
$\text~r$: Abuja is not the capital of Nigeria; false
2.13
1.
True
2.
False
3.
True
2.14
1.
True
2.
False
3.
True
2.15
1.
$p$ $q$ $r$ $\text{~}q$ $\text{~}q \wedge p$ $(\text{~}q \wedge p) \vee r$
T T F F F F
false
2.
$p$ $q$ $r$ $\text{~}r$ $p \vee q$ $(p \vee q) \wedge (\text{~}r)$
T T F T T T
true
3.
$p$ $q$ $r$ $(p \wedge r)$ $\text{~}(p \wedge r)$ $\text{~}(p \wedge r) \wedge q$
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
$p$ $q$ $\text{~}{p}$ ${q} \wedge {p}$ $\text{~}{p} \to \left( {{q} \wedge {p}} \right)$
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
$p$ $q$ ${p} \wedge {q}$ $\text{~}\left( {{p} \wedge {q}} \right)$ $\text{~}{p}$ $\text{~}{q}$ $\text{~}{p} \vee \text{~}{q}$ $\text{~}\left( {{p} \wedge {q}} \right) \leftrightarrow \left( {\text{~}{p} \vee \text{~}{q}} \right)$
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
$p$ $q$ $\text{~}{p}$ ${q} \wedge {p}$ $\text{~}{p} \leftrightarrow \left( {{q} \wedge {p}} \right)$
T T F T F
T F F F T
F T T F F
F F T F F
3.
Valid
$p$ $q$ ${p} \to {q}$ $\text{~}{p}$ $\text{~}{p} \vee {q}$ $\left( {p \to q} \right) \leftrightarrow \left( {\text{~}p \vee q} \right)$
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
$p$ $q$ $r$ $\text{~}{p}$ $\text{~}{q}$ ${p} \wedge {q}$ $\left( {{p} \wedge {q}} \right) \to {r}$ $\text{~}{p} \vee \text{~}{q}$ $\left( {{\text{~}p} \vee {\text{~}q}} \right) \vee {r}$ $\left( {{p} \wedge {q} \to {r}} \right) \leftrightarrow \left( {{\text{~}p} \vee {\text{~}q} \vee {r}} \right)$
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 $\text{~}q \to\text{~}p.$
${p}$ ${q}$ ${p} \to {q}$ $\text{~}{q}$ $\text{~}{p}$ $\text{~}{q} \to \text{~}{p}$ $\left( {p} \to {q} \right) \leftrightarrow \left( {\text{~}{q} \to \text{~}{p}} \right)$
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 $p \vee \text{~}q.$
$p$ $q$ ${p} \to {q}$ $\text{~}{q}$ ${p}\rm \vee \text{~}{q}$ $\left( {{p} \to {q}} \right) \leftrightarrow \left( {{p}{\rm{ }} \vee \text{~}{q}} \right)$
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.
$p$ $q$ ${p} \vee {q}$ $\text{~}\left( {{p} \vee {q}} \right)$ $\text{~}p$ $\text{~}q$ $\text{~}p \wedge \text{~}q$ $\text{~}({p} \vee {q}) \leftrightarrow \left( {\text{~}p \wedge \text{~}q} \right)$
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.
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.

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.
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.
$\text{~}p \vee \text{~}q$
31.
$\text{~}p \wedge \text{~}q$
32.
$p \wedge \text{~}q$
33.
De Morgan’s Laws
34.
premise
35.
valid
36.
inductive
37.
deductive
38.
fallacy
39.
sound
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