1.4: Exercises
- 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 \(|A\times B|=mn\text{.}\)