As usual. I mostly care about the problems ## 1.1 Set Theory and Logic ### 1.1.1 Fundamental Concepts 1. Check the distributive laws for $\cup$ and $\cap$ and DeMorgan's laws. 2. Determine which of the following statements are true for all sets $A, B, C$, and $D$. If a double implication fails, determine whether one or the other of the possible implications holds. If an equality fails, determine whether the statement becomes true if the "equals" symbol is replaced by one or the other of the inclusion symbols $\subset$ or $\supset$. - (a) $A \subset B$ and $A \subset C \Leftrightarrow A \subset(B \cup C)$ - (b) $A \subset B$ or $A \subset C \Leftrightarrow A \subset(B \cup C)$ - (c) $A \subset B$ and $A \subset C \Leftrightarrow A \subset(B \cap C)$ - (d) $A \subset B$ or $A \subset C \Leftrightarrow A \subset(B \cap C)$ - (e) $A-(A-B)=B$ - (f) $A-(B-A)=A-B$ - (g) $A \cap(B-C)=(A \cap B)-(A \cap C)$ - (h) $A \cup(B-C)=(A \cup B)-(A \cup C)$ - (i) $(A \cap B) \cup(A-B)=A$. - (j) $A \subset C$ and $B \subset D \Rightarrow(A \times B) \subset(C \times D)$ - (k) The converse of (j). - (l) The converse of $(\mathrm{j})$, assuming that $A$ and $B$ are nonempty. - (m) $(A \times B) \cup(C \times D)=(A \cup C) \times(B \cup D)$ - (n) $(A \times B) \cap(C \times D)=(A \cap C) \times(B \cap D)$ - (o) $A \times(B-C)=(A \times B)-(A \times C)$ - (p) $(A-B) \times(C-D)=(A \times C-B \times C)-A \times D$ - (q) $(A \times B)-(C \times D)=(A-C) \times(B-D)$ 3. Write the contrapositive and converse of the following statement: "If $x<0$, then $x^{2}-x>0$," and determine which (if any) of the three statements are true. - Do the same for the statement "If $x>0$, then $x^{2}-x>0$." 4. Let $A$ and $B$ be sets of real numbers. Write the negation of each of the following statements: - (a) For every $a \in A$, it is true that $a^{2} \in B$ - (b) For at least one $a \in A$, it is true that $a^{2} \in B$ - (c) For every $a \in A$, it is true that $a^{2} \notin B$ - (d) For at least one $a \notin A$, it is true that $a^{2} \in B$. 5. Let $\mathcal{A}$ be a nonempty collection of sets. Determine the truth of each of the following statements and of their converses: - (a) $x \in \bigcup_{A \in \mathcal{A}} A \Rightarrow x \in A$ for at least one $A \in \mathcal{A}$. - (b) $x \in \bigcup_{A \in \mathcal{A}} A \Rightarrow x \in A$ for every $A \in \mathcal{A}$. - (c) $x \in \bigcap_{A \in \mathcal{A}} A \Rightarrow x \in A$ for at least one $A \in \mathcal{A}$. - (d) $x \in \bigcap_{A \in \mathcal{A}} A \Rightarrow x \in A$ for every $A \in \mathcal{A}$. 6. Write the contrapositive of each of the statements of Exercise 5. 7. Given sets $A, B$, and $C$, express each of the following sets in terms of $A, B$, and $C$, using the symbols $\cup, \cap$, and $-$.$\begin{aligned} D &=\{x \mid x \in A \text { and }(x \in B \text { or } x \in C)\} \\ E &=\{x \mid(x \in A \text { and } x \in B) \text { or } x \in C\} \\ F &=\{x \mid x \in A \text { and }(x \in B \Rightarrow x \in C)\} \end{aligned}$ 8. If a set $A$ has two elements, show that $\mathcal{P}(A)$ has four elements. How many elements does $\mathcal{P}(A)$ have if $A$ has one element? Three elements? No elements? Why is $\mathcal{P}(A)$ called the power set of $A$? 9. Formulate and prove DeMorgan's laws for arbitrary unions and intersections. 10. Let $\mathbb{R}$ denote the set of real numbers. For each of the following subsets of $\mathbb{R} \times \mathbb{R}$, determine whether it is equal to the cartesian product of two subsets of $\mathbb{R}$. - (a) $\{(x, y) \mid x$ is an integer $\}$ - (b) $\{(x, y) \mid 0<y \leq 1\}$ - (c) $\{(x, y) \mid y>x\}$ - (d) $\{(x, y) \mid x\notin \mathbb{Z},\ y\in\mathbb{Z}\}$ - (e) $\left\{(x, y) \mid x^{2}+y^{2}<1\right\}$ ### 1.1.2 Functions ### 1.1.3 Relations ### 1.1.4 The Integers and the Real Numbers ### 1.1.5 Cartesian Products ### 1.1.6 Finite Sets ### 1.1.7 Countable and Uncountable Sets ### 1.1.8 The Principle of Recursive Definition ### 1.1.9 Infinite Sets and the Axiom of Choice ### 1.1.10 Well-Ordered Sets ### 1.1.11 The Maximum Principle Supplementary Exercises: Well-Ordering Chapter 2 Topological Spaces and Continuous Functions 12 Topological Spaces 13 Basis for a Topology 14 The Order Topology 15 The Product Topology on X × Y 16 The Subspace Topology 17 Closed Sets and Limit Points 18 Continuous Functions 19 The Product Topology 20 The Metric Topology 21 The Metric Topology (continued) 22 The Quotient Topology Supplementary Exercises: Topological Groups Chapter 3 Connectedness and Compactness 23 Connected Spaces 24 Connected Subspaces of the Real Line 25 Components and Local Connectedness 26 Compact Spaces 27 Compact Subspaces of the Real Line 28 Limit Point Compactness 29 Local Compactness Supplementary Exercises: Nets