1.4: Exercises

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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.)

What are \(f\)'s domain, codomain, and range?
Prove or disprove each of the following statements. (Your proofs do not need to be long to be correct!)
1. \(f\) is onto;
2. \(f\) is 1-1;
3. \(f\) is a bijection. (You may refer to parts (i) and (ii) for this part.)
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.)
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 \(|A\times B|=mn\text{.}\)