5.9.5: Chapter Review
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Statements and Quantifiers
Fill in the blanks to complete the following sentences.
Exercise \(\PageIndex{1}\)
The ______________ of a logical statement has the opposite truth value of the original statement.
Exercise \(\PageIndex{2}\)
_______________ are logical statements presented as the facts used to support the conclusion of a logical argument.
Determine whether each of the following sentences represents a logical statement, also called a proposition. If it is a logical statement, determine whether it is true or false.
Exercise \(\PageIndex{3}\)
Where is the restroom?
Exercise \(\PageIndex{4}\)
No even numbers are odd numbers.
Exercise \(\PageIndex{5}\)
4 + 3 = 8.
Write the negation of each following statement symbolically and in words.
Exercise \(\PageIndex{6}\)
\(\text{~}p{:}\) Pink Floyd’s album The Wall is not a rock opera.
Exercise \(\PageIndex{7}\)
\(q{:}\) Some dogs are Labrador retrievers.
Exercise \(\PageIndex{8}\)
\(\text{~}r{:}\) Some universities are not expensive.
Draw a logical conclusion to the following arguments, and include in both one of the following quantifiers: all, some, or none.
Exercise \(\PageIndex{9}\)
Spaghetti noodles are made with wheat, ramen noodles are made with wheat, and lo mein noodles are made with wheat.
Exercise \(\PageIndex{10}\)
A Porsche Boxster does not have four doors, a Volkswagen Beetle does not have four doors, and a Mazda Miata does not have four doors.
Compound Statements
Fill in the blanks to complete the following sentences.
Exercise \(\PageIndex{11}\)
___________________ are words or symbols used to join two or more logical statement together to from a compound statement.
Exercise \(\PageIndex{12}\)
__________________ and __________________ have equal dominance and are evaluated from left to right when no parentheses are present in a compound logical statement.
Translate each compound statement below into symbolic form. Given: \(p{:}\) “Tweety Bird is a bird,” \(q{:}\) “Bugs is a bunny,” \(r{:}\) “Bugs says, ‘What’s up, Doc?’,” \(s{:}\) “Sylvester is a cat,” and \(t{:}\) “Sylvester chases Tweety Bird.”
Exercise \(\PageIndex{13}\)
If Tweety Bird is a bird, then Sylvester will not chase him.
Exercise \(\PageIndex{14}\)
Tweety Bird is a bird and Sylvester chases him if and only if Bugs says, “What’s up Doc?”
Translate the symbolic form of each compound logical statement below into words.
Given: \(p{:}\) “Tweety Bird is a bird,” \(q{:}\) “Bugs is a bunny,” \(r{:}\) “Bugs says, ‘What’s up, Doc?’,” \(s{:}\) “Sylvester is a cat,” and \(t{:}\) “Sylvester chases Tweety Bird.”
Exercise \(\PageIndex{15}\)
\(\sim q \vee p \rightarrow \sim s\)
Exercise \(\PageIndex{16}\)
\(\sim(p \wedge \sim s) \leftrightarrow q \rightarrow r\)
For each of the following compound logical statements, apply the proper dominance of connectives by adding parentheses to indicate the order to evaluate the statement.
Exercise \(\PageIndex{17}\)
\(p \vee q \wedge r \rightarrow \sim s \wedge t\)
Exercise \(\PageIndex{18}\)
\(\sim p \rightarrow q \vee r \leftrightarrow p \wedge s \rightarrow \sim t\)
Constructing Truth Tables
Fill in the blanks to complete the sentences.
Exercise \(\PageIndex{19}\)
A ______________ is true if at least one of its component statements is true.
Exercise \(\PageIndex{20}\)
For a ____________________ to be true, all of its component statements must be true.
Given the statements, \(p{:}\) “No fish are mammals,” \(q{:}\) “All lions are cats,” and \(\text{~}r{:}\) “Some birds do not lay eggs,” construct a truth table to determine the truth value of each compound statement below.
Exercise \(\PageIndex{21}\)
\(p \wedge \sim r\)
Exercise \(\PageIndex{22}\)
\(\sim(p \vee \sim r)\)
Exercise \(\PageIndex{23}\)
\(\sim p \vee q \wedge \sim(\sim r)\)
Construct a truth table to analyze all the possible outcomes of the following statements, and determine whether the statements are valid.
Exercise \(\PageIndex{24}\)
\(\sim p \vee q \wedge p\)
Exercise \(\PageIndex{25}\)
\(\sim p \vee q \vee \sim q\)
Truth Tables for the Conditional and Biconditional
Fill in the blanks to complete the following sentences.
Exercise \(\PageIndex{26}\)
If the , \(p\), of a conditional statement is true, then the conclusion, \(q\), must also be true for the conditional statement \(p \rightarrow q\) to be true.
Exercise \(\PageIndex{27}\)
The biconditional statement \(p \leftrightarrow q\) is whenever the truth value of \(p\) matches the truth value of \(q\), otherwise it is
Complete the truth tables below to determine the truth value of the proposition in the last column.
Exercise \(\PageIndex{28}\)
| \(p\) | \(q\) | \(r\) | \(p \vee q\) | \( \sim(p \vee q) \) | \(\text{~}{r}\) | \(\sim(p \vee q) \rightarrow \sim r\) |
|---|---|---|---|---|---|---|
| F | F | T |
Exercise \(\PageIndex{29}\)
| \(p\) | \(q\) | \(\text{~}{q}\) | \(p \rightarrow q \) | \(\sim(p \rightarrow q)\) | \(p \wedge \sim q \) | \(\sim(p \rightarrow q) \leftrightarrow(p \wedge \sim q)\) |
|---|---|---|---|---|---|---|
| T | F |
Assume the following statements are true. \(p{:}\) “Poof is a baby fairy,” \(q{:}\) “Timmy Turner has fairly odd parents,” \(r{:}\) “Cosmo and Wanda will grant Timmy’s wishes,” and \(t{:}\) “Timmy Turner is 10 years old.” Translate each of the following statements into symbolic form, then determine its truth value.
Exercise \(\PageIndex{30}\)
If Timmy Turner is 10 years old and Poof is not a baby fairy, then Timmy Turner has fairly odd parents.
Exercise \(\PageIndex{31}\)
Cosmos and Wanda will not grant Timmy’s wishes if and only if Timmy Turner is 10 years old or he does not have fairly odd parents.
Exercise \(\PageIndex{32}\)
Construct a truth table to analyze all the possible outcomes and determine the validity of the following argument.
\(\text{~}p \vee q{\text{ }} \leftrightarrow {\text{ }}\text{~}q \to {\text{ }}\text{~}p\)
Equivalent Statements
Fill in the blanks to complete the sentences below. 33. The _________________ is logically equivalent to the inverse \( \sim p \rightarrow \sim q \)\(\text{~}p \to\text{~}q.\)
Exercise \(\PageIndex{34}\)
The _________________ is logically equivalent to the conditional \(p \rightarrow q \)\(p \to q.\)
Use the conditional statement, \(p \rightarrow q \)\(p \to q{:}\) “If Novak makes the basket, then Novak’s team will win the game," to answer the following questions. 35. Write the conclusion of the conditional statement in words and label it appropriately.
Exercise \(\PageIndex{36}\)
Write the hypothesis of the conditional statement in words and label it appropriately.
Exercise \(\PageIndex{37}\)
Identify the following statement as the converse, inverse, or contrapositive: “If Novak does not make the basket, then his team will not win the game.”
Exercise \(\PageIndex{38}\)
Identify the following statement as the converse, inverse, or contrapositive: “If Novak’s team wins the game, then he made the basket.”
De Morgan’s Laws
Fill in the blanks to complete the sentences.
Exercise \(\PageIndex{39}\)
De Morgan’s Law for the negation of a disjunction states that \( \(\text{~}(p \vee q) \equiv \)____________.
Exercise \(\PageIndex{40}\)
De Morgan’s Law for the negation of a conjunctions states that _____________\( \(\equiv {\text{ }}\text{~}p{\text{ }} \vee \text{~}q.\)
Exercise \(\PageIndex{41}\)
Apply De Morgan’s Law to write the statement without parentheses: \( \(\text{~}\left( {\text{~}p \wedge q} \right)\).
Exercise \(\PageIndex{42}\)
Apply the property for the negation of a conditional to write the statement as a conjunction or disjunction: \( \(\text{~}(\text{~}p \wedge q \to\text{~}r)\).
Exercise \(\PageIndex{43}\)
Write the negation of the conditional statement in words: If Thomas Edison invented the phonograph, then albums are made of vinyl, or the transistor radio was the first portable music device.
Exercise \(\PageIndex{44}\)
Construct a truth table to verify that the logical property is valid: \(\text{~}\left( {\text{~}p \to\text{~}q} \right) \equiv \text{~}p \wedge q\).
Logical Arguments
Fill in the blanks to complete the sentences below.
Exercise \(\PageIndex{45}\)
The _____ ___ __________ is a valid logical argument with premises, \(p \to q\) and \(p\), used to support the conclusion, \(q.\)
Exercise \(\PageIndex{46}\)
The chain rule for conditional arguments states that the ___________________ property applies to conditional arguments, so that: \( \(\left( {p \to q} \right) \wedge \left( {q \to r} \right) \to \left( {p \to r} \right).\)
Assume each pair of statements represents true premises in a logical argument. Based on these premises, state a valid conclusion that is consistent with the form of the argument.
Exercise \(\PageIndex{47}\)
If the Tampa Bay Buccaneers did not win Super Bowl LV, then Tom Brady was not their quarterback. Tom Brady was the Tampa Bay Buccaneers quarterback.
Exercise \(\PageIndex{48}\)
If \(\text{~}q\), then \(p\) and if \(r\), then \(\text{~}q\).
Exercise \(\PageIndex{49}\)
If Kamala Harris is the vice president of the United States, then Kamala Harris is the president of the U.S. Senate. Kamala Harris is the vice president of the United States.
Exercise \(\PageIndex{50}\)
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 all frogs are brown, then Kermit is not a frog. Kermit is a frog. Therefore, some frogs are not brown.