Contemporary Mathematics

# Chapter Test

### Chapter Test

Determine whether each of the following sentences represent a logical statement. If it is a logical statement, determine whether it is true or false.
1 .
$1 + 2 - 3 = 0$
2 .
3 .
All mammals lay eggs.

Write the negation of each statement below.

4 .
Some monkeys do not have tails.
5 .
$p \wedge \text{~}q$
6 .
If the plumber does not remove the clog, then the homeowner will not pay the plumber.
Given: $p{:}$ Frodo is a hobbit, $q{:}$ Gandalf is a wizard, $r{:}$ Frodo and Samwise will take the ring to Mordor, and $s{:}$ Gollum will help Frodo get into Mordor.
Translate the symbolic form of each compound logical statement into words.
7 .
$p \vee q \leftrightarrow r$
8 .
$\text{~}\left( {\text{~}r \vee \text{~}s} \right)$
Translate the written form of each compound logical statement into symbolic form.
9 .
Frodo and Samwise will take the ring to Mordor or Gandalf is not a wizard and Frodo is a hobbit.
10 .
If Gollum will not help Frodo get into Mordor, then Gandalf is not a wizard and Frodo is not a hobbit.
For each of the following compound logical statements, apply the proper dominance of connectives by adding parentheses to indicate the order in which the statement must be evaluated.
11 .
$\text{~}p \to q \leftrightarrow r$
12 .
$\text{~}\left( {p \wedge q} \right) \leftrightarrow {\text{ }}\text{~}p \vee \text{~}q$
13 .
Complete the truth table to determine the truth value of the proposition in the last column.
$p$ $q$ ${p} \wedge {q}$ $\text{~}{q}$ ${p}{\text{ }} \vee \text{~}{q}$ $\left( {{p} \wedge {q}} \right) \leftrightarrow \left( {{p}{\text{ }} \vee \text{~}{q}} \right)$
F T
Given the true statements $p{:}$ “A right triangle has one 90-degree angle," $q{:}$ "The triangle is a right triangle," $r{:}$ "${a^2} + {b^2} = {c^2},$" and $s{:}$ "The longest side of a triangle is $c$ implies $a + b$ must be $> c$." Write each of the following compound statements in symbolic form, then construct a truth table to determine the truth value of the compound statement.
14 .
If a triangle is a right triangle, then it does not have one 90-degree angle or ${a^2} + {b^2} = {c^2}.$
15 .
The triangle is a right triangle, or a right triangle does not have a 90-degree angle, if and only if it is not the case that the longest side of a triangle is $c$ implies $a + b$ must be $> c$.
Use the conditional statement, $p \to q{:}$ "If Phil Mickelson is 50 years old, then Phil Mickelson won the Player’s Championship," to answer the following questions.
16 .
Write the converse statement in words.
17 .
If the conditional statement is true, and the hypothesis is true, what is a valid conclusion to the argument?
18 .
If the conditional statement is true, and the conclusion is false, what is a valid conclusion to the argument?
19 .
Construct a truth table to analyze all the possible outcomes and determine the validity of the following argument.
$\text{~}p \vee q \leftrightarrow q \to p$
20 .
Construct a truth table or Venn diagram to prove whether the following argument is valid. If the argument is valid, determine whether it is sound.
If John Mayer played MTV unplugged, then some guitars are acoustic. John Mayer played MTV unplugged. Therefore, some guitars are acoustic.
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