91Ó°ÊÓ

Let \(A\) be a subset of some universal set \(U\). Prove each of the following (from Theorem 5.20): (a) \( \left(A^{c}\right)^{c}=A\) \((\mathbf{c}) \emptyset^{c}=U\) (b) \(A-\emptyset=A\) (d) \(U^{c}=\emptyset\)

Sketch a graph of each of the following Cartesian products in the Cartesian plane. (a) \([0,2] \times[1,3]\) (e) \(\mathbb{R} \times(2,4)\) (b) \((0.2) \times(1,3]\) (f) \((2,4) \times \mathbb{R}\) (c) \([2,3] \times\\{1\\}\) (g) \(\mathbb{R} \times\\{-1\\}\) (d) \(\\{1\\} \times[2,3]\) (h) \(\\{-1\\} \times[1,+\infty)\)

Let \(A, B,\) and \(C\) be subsets of a universal set \(U\). (a) Draw a Venn diagram with \(A \subseteq B\) and \(B \subseteq C .\) Does it appear that \(A \subseteq C ?\) (b) Prove the following proposition: If \(A \subseteq B\) and \(B \subseteq C,\) then \(A \subseteq C\) This may seem like an obvious result. However, one of the reasons for this exercise is to provide practice at properly writing a proof that one set is a subset of another set. So we should start the proof by assuming that \(A \subseteq B\) and \(B \subseteq C .\) Then we should choose an arbitrary element of \(A\).

For each natural number \(n,\) let \(A_{n}=\\{k \in \mathbb{N} \mid k \geq n\\} .\) Assuming the universal set is \(\mathbb{N}\), use the roster method or set builder notation to specify each of the following sets: (a) \(\bigcap_{j=1}^{5} A_{j}\) (e) \(\bigcup_{j=1}^{5} A_{j}\) (b) \(\left(\bigcap_{j=1}^{5} A_{j}\right)^{c}\) $$ \text { (f) }\left(\bigcup_{j=1}^{5} A_{j}\right)^{c} $$ (c) \(\bigcap_{j=1}^{5} A_{j}^{c}\) (g) \(\bigcap_{j \in \mathbb{N}} A_{j}\) (d) \(\bigcup_{j=1}^{5} A_{j}^{c}\) (h) \(\bigcup_{j \in \mathbb{N}} A_{j}\)

For each positive real number \(r,\) define \(T_{r}\) to be the closed interval \(\left[-r^{2}, r^{2}\right]\) That is, \(T_{r}=\left\\{x \in \mathbb{R} \mid-r^{2} \leq x \leq r^{2}\right\\}\) Let \(\Lambda=\\{m \in \mathbb{N} \mid 1 \leq m \leq 10\\}\). Use either interval notation or set builder notation to specify each of the following sets: * (a) \(\bigcup_{k \in \Lambda} T_{k}\) (c) \(\bigcup_{r \in \mathbb{R}^{+}} T_{r}\) (e) \(\bigcup_{k \in \mathbb{N}} T_{k}\) *(b) \(\bigcap_{k \in \Lambda} T_{k}\) (d) \(\bigcap_{r \in \mathbb{R}^{+}} T_{r}\) (f) \(\bigcap_{k \in \mathbb{N}} T_{k}\)

Prove Theorem 5.25, Part (4): \((A \cup B) \times C=(A \times C) \cup(B \times C)\).

Let \(C=\\{x \in \mathbb{Z} \mid x \equiv 7(\bmod 9)\\}\) and \(D=\\{x \in \mathbb{Z} \mid x \equiv 1(\bmod 3)\\}\). (a) List at least five different elements of the set \(C\) and at least five elements of the set \(D\). (b) Is \(C \subseteq D\) ? Justify your conclusion with a proof or a counterexample. (c) Is \(D \subseteq C\) ? Justify your conclusion with a proof or a counterexample.

Write all of the proper subset relations that are possible using the sets of numbers \(\mathbb{N}, \mathbb{Z}, \mathbb{Q},\) and \(\mathbb{R}\).

Let \(A, B,\) and \(C\) be subsets of some universal set \(U\) (a) Draw two general Venn diagrams for the sets \(A, B,\) and \(C .\) On one, shade the region that represents \(A-(B \cup C),\) and on the other, shade the region that represents \((A-B) \cap(A-C) .\) Based on the Venn diagrams, make a conjecture about the relationship between the sets \(A-(B \cup C)\) and \((A-B) \cap(A-C)\) (b) Use the choose-an-element method to prove the conjecture from Exercise (4a). (c) Use the algebra of sets to prove the conjecture from Exercise (4a).

Let \(A, B,\) and \(C\) be subsets of some universal set \(U\) (a) Draw two general Venn diagrams for the sets \(A, B,\) and \(C .\) On one, shade the region that represents \(A-(B \cap C),\) and on the other, shade the region that represents \((A-B) \cup(A-C) .\) Based on the Venn diagrams, make a conjecture about the relationship between the sets \(A-(B \cap C)\) and \((A-B) \cup(A-C)\) (b) Use the choose-an-element method to prove the conjecture from Exercise \((5 a)\) (c) Use the algebra of sets to prove the conjecture from Exercise (5a).