2.5.0: Exercises
- Page ID
- 171693
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)For the following exercises, determine whether the pair of compound statements are logically equivalent by constructing a truth table.
Converse: \(q \rightarrow p\) and inverse: \(\sim p \rightarrow \sim q\)
Conditional: \(p \rightarrow q\) and contrapositive: \(\sim q \rightarrow \sim p\)
Inverse: \(\sim p \rightarrow \sim q\) and contrapositive: \(\sim q \rightarrow \sim p\)
Conditional: \(p \rightarrow q\) and converse: \(q \rightarrow p\)
\(\sim p \rightarrow q\) and \(p \vee \sim q\)
\(\sim p \rightarrow q\) and \(p \vee q\)
\(\sim(p \wedge q)\) and \(\sim p \wedge \sim q\)
\(\sim(p \wedge q)\) and \(\sim p \vee \sim q\)
\(p \wedge(q \vee r)\) and \((p \wedge q) \vee(p \wedge r)\)
\(p \wedge(q \vee r)\) and \((p \wedge q) \vee r\)
- Write the conditional statement \(p \rightarrow q\) in words.
- Write the converse statement \(q \rightarrow p\) in words.
- Write the inverse statement \(\sim p \rightarrow \sim q\) in words.
- Write the contrapositive statement \(\sim q \rightarrow \sim p\) in words.
\(p\) : Six is afraid of Seven and \(q\) : Seven ate Nine.
\(p\) : Hope is eternal and \(q\) : Despair is temporary.
\(p\) : Tom Brady is a quarterback and \(q\) : Tom Brady does not play soccer.
\(p\) : Shakira does not sing opera and \(q\) : Shakira sings popular music.
\(p\) :The shape does not have three sides and \(q\) : The shape is not a triangle.
\(p\) : All birds can fly and \(q\) : Emus can fly.
\(p\) : Penguins cannot fly and \(q\) : Some birds can fly.
\(p\) : Some superheroes do not wear capes and \(q\) : Spiderman is a superhero.
\(p\) : No Pokémon are little ponies and \(q\) : Bulbasaur is a Pokémon.
\(p\) : Roses are red, and violets are blue and \(q\) : Sugar is sweet, and you are sweet too.
Write the hypothesis of the conditional statement, label it with a \(p\), and determine its truth value.
Write the conclusion of the conditional statement, label it with a \(q\), and determine its truth value.
Identify the following statement as the converse, inverse, or contrapositive, and determine its truth value: "If Clark Kent is not Superman, then Lois Lane is a reporter."
Identify the following statement as the converse, inverse, or contrapositive, and determine its truth value: "If Lois Lane is a reporter, then Clark Kent is not Superman."
Which form of the conditional is logically equivalent to the converse?
Write the hypothesis of the conditional statement, label it with a \(p\), and determine its truth value.
Write the conclusion of the conditional statement, label it with a \(q\), and determine its truth value.
Identify the following statement as the converse, inverse, or contrapositive, and determine its truth value: "If Donnie Wahlberg was a member of New Kids on the Block, then The Masked Singer is not a music competition."
Identify the following statement as the converse, inverse, or contrapositive, and determine its truth value: "If The Masked Singer is a music competition, then Donnie Wahlberg was not a member of New Kids on the Block."
Which form of the conditional is logically equivalent to the contrapositive, \(\sim q \rightarrow \sim p\) ?
Write the hypothesis of the conditional statement, label it with a \(p\), and determine its truth value.
Write the conclusion of the conditional statement, label it with a \(q\), and determine its truth value.
Identify the following statement as the converse, inverse, or contrapositive, and determine its truth value: "If some fish are whales, then some whales are not mammals."
Write the inverse in words and determine its truth value.
Write the converse in words and determine its truth value.
Write the hypothesis of the conditional statement, label it with a \(p\), and determine its truth value.
Write the conclusion of the conditional statement, label it with a \(q\), and determine its truth value.
Write the converse in words and determine its truth value.
Write the contrapositive in words and determine its truth value.
Write the inverse in words and determine its truth value.