2.6.0: Exercises
- Page ID
- 171694
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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, use De Morgan's Laws to write each statement without parentheses.
\(\sim(\sim p \vee q)\)
\(\sim(\sim p \wedge \sim q)\)
\(\sim(p \wedge \sim q)\)
\(\sim(\sim p \vee \sim q)\)
For the following exercises, use De Morgan's Laws to write the negation of each statement in words without using the phrase, "It is not the case that, ..."
Sergei plays right wing and Patrick plays goalie.
Mario is a carpenter, or he is a plumber.
Luigi is a plumber, or he is not a video game character.
Ralph Macchio was the original Karate Kid, and karate is not for defense only.
Some people like broccoli, but my siblings did not like broccoli.
Some people do not like chocolate or all people like pizza.
For the following exercises, write each statement as a conjunction or disjunction in symbolic form by applying the property for the negation of a conditional.
\(\sim(p \rightarrow q)\)
\(\sim(p \rightarrow \sim q)\)
\(\sim(\sim p \rightarrow q)\)
\(\sim(\sim p \rightarrow \sim q)\)
\(\sim(p \wedge q \rightarrow \sim r)\)
\(\sim(p \rightarrow q \vee r)\)
\(\sim(p \vee q \rightarrow \sim r)\)
\(~(p \rightarrow q \wedge r)\)
For the following exercises, write the negation of each conditional in words by applying the property for the negation of a conditional.
If a student scores an 85 on the final exam, then they will receive an \(A\) in the class.
If a person does not pass their road test, then they will not receive their driver's license.
If a student does not do their homework, then they will not play video games.
If a commuter misses the bus, then they will not go to work today.
If a racecar driver gets pulled over for speeding, then they will not make it to the track on time for the race.
If Rene Descartes was a philosopher, then he was not a mathematician.
If George Boole invented Boolean algebra and Thomas Edison invented the light bulb, then Pacman is not the best video game ever.
If Jonas Salk created the polio vaccine, then his child received the vaccine or his child had polio.
If Billie Holiday sang the blues or Cindy Lauper sang about true colors, then John Lennon was not a Beatle.
If Percy Jackson is the lightning thief and Artemis Fowl is a detective, then Artemis Fowl will catch Percy Jackson.
If all rock stars are men, then Pat Benatar is not a rock star.
If Lady Gaga is a rock star, then some rock stars are women.
If yellow combined with blue makes green, then all colors are beautiful.
If leopards have spots and zebras have stripes, then some animals are not monotone in color. For the following exercises, construct a truth table to verify that the logical property is valid.
\(p \rightarrow q \equiv \sim p \vee q\)
\(p \rightarrow \sim q \equiv \sim p \vee \sim q\)
\(\sim p \rightarrow q \equiv p \vee q\)
\(\sim(p \rightarrow \sim q \vee \sim r) \equiv p \wedge q \wedge r\)