1.4: Exercises
- Page ID
- 84799
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)- Yes/No. For each of the following, write Y if the object described is a well-defined set; otherwise, write N. You do NOT need to provide explanations or show work for this problem.
- \(\{z \in \mathbb{C} \,:\, |z|=1\}\)
- \(\{\epsilon \in \mathbb{R}^+\,:\, \epsilon \mbox{ is sufficiently small} \}\)
- \(\{q\in \mathbb{Q} \,:\, q \mbox{ can be written with denominator } 4\}\)
- \(\{n \in \mathbb{Z}\,:\, n^2 \lt 0\}\)
- List the elements in the following sets, writing your answers as sets.
Example: \(\{z\in \mathbb{C}\,:\,z^4=1\}\) Solution: \(\{\pm 1, \pm i\}\)
- \(\{z\in \mathbb{R}\,:\, z^2=5\}\)
- \(\{m \in \mathbb{Z}\,:\, mn=50 \mbox{ for some } n\in \mathbb{Z}\}\)
- \(\{a,b,c\}\times \{1,d\}\)
- \(P(\{a,b,c\})\)
- Let \(S\) be a set with cardinality \(n\in \mathbb{N}\text{.}\) Use the cardinalities of \(P(\{a,b\})\) and \(P(\{a,b,c\})\) to make a conjecture about the cardinality of \(P(S)\text{.}\) You do not need to prove that your conjecture is correct (but you should try to ensure it is correct).
4. Let \(f: \mathbb{Z}^2 \to \mathbb{R}\) be defined by \(f(a,b)=ab\text{.}\) (Note: technically, we should write \(f((a,b))\text{,}\) not \(f(a,b)\text{,}\) since \(f\) is being applied to an ordered pair, but this is one of those cases in which mathematicians abuse notation in the interest of concision.)
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What are \(f\)'s domain, codomain, and range?
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Prove or disprove each of the following statements. (Your proofs do not need to be long to be correct!)
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\(f\) is onto;
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\(f\) is 1-1;
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\(f\) is a bijection. (You may refer to parts (i) and (ii) for this part.)
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Find the images of the element \((6,-2)\) and of the set \(\mathbb{Z}^- \times \mathbb{Z}^-\) under \(f\text{.}\) (Remember that the image of an element is an element, and the image of a set is a set.)
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Find the preimage of \(\{2,3\}\) under \(f\text{.}\) (Remember that the preimage of a set is a set.)
- Let \(S\text{,}\) \(T\text{,}\) and \(U\) be sets, and let \(f: S\to T\) and \(g: T\to U\) be onto. Prove that \(g \circ f\) is onto.
- Let \(A \) and \(B\) be sets with \(|A|=m\lt \infty\) and \(|B|=n\lt \infty\text{.}\) Prove that \(|A\times B|=mn\text{.}\)