2.4: Activities
- Page ID
- 83406
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)Activity \(\PageIndex{1}\)
Write an English language statement that has the logical form \(\neg (A \lor B)\text{.}\) Then write one that has the form \(\neg A \land \neg B\text{,}\) where \(A\) and \(B\) are the same as in your first sentence. DeMorgan's Laws say your two sentences are logically equivalent. Do you agree?
Activity \(\PageIndex{2}\)
What do you think DeMorgan's Laws would say about \(\neg (A \land B \land C)\text{?}\) Use propositional calculus to justify your answer.
Activity \(\PageIndex{3}\)
Recall that a pair of coordinates \((x,y)\) defines a point in the Cartesian plane.
Consider the following conditional statement.
If Cartesian points \((a,b)\) and \((c,d)\) are actually the same point, then \(a = c\text{.}\)
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Write out the converse, inverse, and contrapositive of the above statement.
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You now have four conditional statements. For each of the four, decide whether it is true, and justify your answer.
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For each of the three new conditional statements from Task a in turn, take the view that that statement is the original conditional, and decide which of the others are its converse, inverse, and contrapositive.
Activity \(\PageIndex{4}\)
In this activity, we will justify the equivalence
\begin{equation*} p \leftrightarrow q \Leftrightarrow(p\rightarrow q) \land (q \rightarrow p) \text{.} \end{equation*}
So consider the statements \(A = p \leftrightarrow q\) and \(B = (p\rightarrow q) \land (q \rightarrow p)\text{.}\)
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Argue that if \(A\) is false, then so is \(B\text{.}\)
Do not use the proposed equivalence above as part of your argument.
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Argue that if \(B\) is false, then so is \(A\text{.}\)
Do not use the proposed equivalence above as part of your argument.
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Explain why the the two arguments in Task a and Task b, taken together, justify the equivalence \(A \Leftrightarrow B\text{.}\) Do this without making any further arguments about the truth values of \(p\) and \(q\text{.}\)
Activity \(\PageIndex{5}\)
Consider the statements \(p \rightarrow (q_1 \lor q_2)\) and \((p \land \neg q_1) \rightarrow q_2\text{.}\)
Use propositional calculus and substitution to show that these two statements are equivalent.