2.4.0: Exercises
- Page ID
- 171692
\( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)
\( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)
\( \newcommand{\dsum}{\displaystyle\sum\limits} \)
\( \newcommand{\dint}{\displaystyle\int\limits} \)
\( \newcommand{\dlim}{\displaystyle\lim\limits} \)
\( \newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\)
( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\)
\( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\)
\( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\)
\( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\)
\( \newcommand{\Span}{\mathrm{span}}\)
\( \newcommand{\id}{\mathrm{id}}\)
\( \newcommand{\Span}{\mathrm{span}}\)
\( \newcommand{\kernel}{\mathrm{null}\,}\)
\( \newcommand{\range}{\mathrm{range}\,}\)
\( \newcommand{\RealPart}{\mathrm{Re}}\)
\( \newcommand{\ImaginaryPart}{\mathrm{Im}}\)
\( \newcommand{\Argument}{\mathrm{Arg}}\)
\( \newcommand{\norm}[1]{\| #1 \|}\)
\( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\)
\( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\AA}{\unicode[.8,0]{x212B}}\)
\( \newcommand{\vectorA}[1]{\vec{#1}} % arrow\)
\( \newcommand{\vectorAt}[1]{\vec{\text{#1}}} % arrow\)
\( \newcommand{\vectorB}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)
\( \newcommand{\vectorC}[1]{\textbf{#1}} \)
\( \newcommand{\vectorD}[1]{\overrightarrow{#1}} \)
\( \newcommand{\vectorDt}[1]{\overrightarrow{\text{#1}}} \)
\( \newcommand{\vectE}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash{\mathbf {#1}}}} \)
\( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)
\(\newcommand{\longvect}{\overrightarrow}\)
\( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)
\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)For the following exercises, complete the truth table to determine the truth value of the proposition in the last column.
| \(p\) | \(q\) | \(\text{~}p\) | \(\text{~}p \to q\) |
|---|---|---|---|
| T | T |
| \(p\) | \(q\) | \(\text{~}q\) | \(p \to \text{~}q\) |
|---|---|---|---|
| T | T |
| \(p\) | \(q\) | \(\text{~}p\) | \(\text{~}p \leftrightarrow q\) |
|---|---|---|---|
| F | T |
| \(p\) | \(q\) | \(\text{~}q\) | \(p \leftrightarrow \text{~}q\) |
|---|---|---|---|
| F | T |
| \(p\) | \(q\) | \(r\) | \(\text{~}p\) | \(\text{~}p \wedge q\) | \((\text{~}p \wedge q) \to r\) |
|---|---|---|---|---|---|
| F | T | F |
| \(p\) | \(q\) | \(r\) | \(\text{~}p\) | \(\text{~}r\) | \(\text{~}p \wedge q\) | \((\text{~}p \wedge q) \to \text{~}r\) |
|---|---|---|---|---|---|---|
| F | F | F |
| \(p\) | \(q\) | \(r\) | \(\text{~}p\) | \(\text{~}r\) | \(\text{~}p \vee q\) | \((\text{~}p \vee q) \leftrightarrow \text{~}r\) |
|---|---|---|---|---|---|---|
| F | F | F |
| \(p\) | \(q\) | \(r\) | \(\text{~}p\) | \(\text{~}r\) | \(\text{~}p \wedge q\) | \((\text{~}p \wedge q) \leftrightarrow \text{~}r\) |
|---|---|---|---|---|---|---|
| T | F | F |
| \(p\) | \(q\) | \(r\) | \(\text{~}p\) | \(\text{~}r\) | \(\text{~}p \vee q\) | \(p \to \text{~}r\) | \((\text{~}p \vee r) \leftrightarrow \left( {p \to \text{~}r} \right)\) |
|---|---|---|---|---|---|---|---|
| F | F | F |
| \(p\) | \(q\) | \(r\) | \(\text{~}p\) | \(\text{~}r\) | \(\text{~}p \wedge q\) | \(p \to \text{~}r\) | \((\text{~}p \wedge q) \leftrightarrow \left( {p \to \text{~}r} \right)\) |
|---|---|---|---|---|---|---|---|
| T | T | T |
For the following exercises, assume these statements are true: \(p{\text{:}}\) Faheem is a software engineer, \(q{\text{:}}\) Ann is a project manager, \(r{\text{:}}\) Giacomo works with Faheem, and \(s{\text{:}}\) The software application was completed on time. Translate each of the following statements to symbols, then construct a truth table to determine its truth value.
If Giacomo works with Faheem, then Faheem is not a software engineer.
If the software application was not completed on time, then Ann is not a project manager.
The software application was completed on time if and only if Giacomo worked with Faheem.
Ann is not a project manager if and only if Faheem is a software engineer.
If the software application was completed on time, then Ann is a project manager, but Faheem is not a software engineer.
If Giacomo works with Faheem and Ann is a project manager, then the software application was completed on time.
The software application was not completed on time if and only if Faheem is a software engineer or Giacomo did not work with Faheem.
Faheem is a software engineer or Ann is not a project manager if and only if Giacomo did not work with Faheem and the software application was completed on time.
Ann is a project manager implies Faheem is a software engineer if and only if the software application was completed on time implies Giacomo worked with Faheem.
If Giacomo did not work with Faheem implies that the software application was not completed on time, then Ann was not the project manager.
For the following exercises, construct a truth table to analyze all the possible outcomes and determine the validity of each argument.
\(p \vee \sim q \rightarrow q\)
\(\sim q \rightarrow p \wedge \sim q\)
\((p \rightarrow q) \leftrightarrow q\)
\((p \rightarrow q) \leftrightarrow p\)
\(\sim(p \vee q) \leftrightarrow(\sim p \wedge \sim q)\)
\((p \rightarrow q) \wedge p \rightarrow q\)
\(p \rightarrow q \rightarrow r\)
\((p \rightarrow q) \wedge(q \rightarrow r) \leftrightarrow(p \rightarrow r)\)
\(p \vee(q \wedge r) \leftrightarrow(p \vee q) \wedge(p \vee r)\)
\(p \vee(q \vee r) \leftrightarrow(p \vee q) \vee r\)