2.4.1: Exercises
- Page ID
- 214376
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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, find the truth value of each statement.
\(p: 7 \times 3=21\). What is the truth value of \(\sim p\) ?
\(q\) : The sun revolves around the Earth. What is the truth value of \(\sim q\) ?
\(\sim r\) : The acceleration of gravity is \(9.81 \mathrm{~m} / \mathrm{sec}^2\). What is the truth value of \(r\) ?
\(s\) : Dan Brown is not the author of the book, The Davinci Code. What is the truth value of \(\sim(\sim s)\) ?
\(t\) : Broccoli is a vegetable. What is the truth value of \(\sim(\sim t)\) ?
For the following exercises, given \(p: 1+2=3, q\) : Five is an even number, and \(r\) : Seven is a prime number, find the truth value of each of the following statements.
\(\sim q\)
\(p \wedge q\)
\(p \vee q\)
\(\sim p \vee \sim q\)
\(p \wedge \sim q\)
\(p \wedge r\)
\(q \wedge r\)
\(q \wedge \sim r\)
\(q \vee \sim r\)
\(\sim(p \wedge r)\)
\(p \vee q \wedge r\)
\(\sim p \vee(q \wedge r)\)
\(\sim(q \wedge r) \vee \sim p\)
\(q \vee r \vee p\)
\(\sim q \wedge r \wedge p\)
For the following exercises, complete the truth table to determine the truth value of the proposition in the last column.
\(p\) |
\(q\) |
 \(r\) |
\(\sim p\) | \(\sim p \vee q\) | \((\sim p \vee q) \wedge r\) |
|---|---|---|---|---|---|
T |
T | T |
| \(p\) |
\(q\) |
\(r\) |
\(\sim p\) | \(\sim p \vee q\) | \((\sim p \vee q) \wedge r\) |
|---|---|---|---|---|---|
| F |
T |
F |
| \(p\) | \(q\) |
\(r\) |
\(\sim p\) | \(\sim r\) | \(\sim p \wedge q(\sim p \wedge q) \vee \sim r\) |
|---|---|---|---|---|---|
| F | F | F |
| \(p\) |
\(q\) |
\(r\) |
\(\sim p\) | \(\sim r\) | \(\sim p \vee q\) | \((\sim p \vee q) \vee \sim r\) |
|---|---|---|---|---|---|---|
| F |
F |
F |
For the following exercises, given \(p\) : All triangles have three sides, \(q\) : Some rectangles are not square, and \(r\) : A pentagon has eight sides, determine the truth value of each compound statement by constructing a truth table.
\(\sim r \wedge q \wedge p\)
\(\sim(q \wedge p) \vee r\)
\(\sim p \vee q \wedge r\)
\(\sim p \vee \sim q \vee r\) For the following exercises, construct a truth table to analyze all the possible outcomes for the following arguments.
\(\sim q \wedge q\)
\(\sim p \vee \sim q\)
\(\sim p \wedge \sim q\)
\(p \wedge q \vee r\) For the following exercises, construct a truth table to determine the validity of each statement.
\(\sim q \vee q\)
\(p \wedge \sim q\)
\(p \wedge q \vee \sim p\)
\((p \wedge q) \vee(\sim p \wedge \sim q)\)