91Ó°ÊÓ

Some English adjectives are descriptive of themselves (for instance, the word polysyllabic is polysyllabic) whereas others are not (for instance, the word monosyllabic is not monosyllabic). The word heterological refers to an adjective that does not describe itself. Is heterological heterological? Explain your answer.

Prove each statement that is true and find a counterexample for each statement that is false. Assume all sets are subsets of a universal set \(U\). For all sets \(A\) and \(B\), if \(A \cap B=\emptyset\) then \(A \times B=\emptyset\).

Indicate which of the following relationships are true and which are false: a. \(\mathbf{Z}^{+} \subseteq \mathbf{Q}\) b. \(\mathbf{R}^{-} \subseteq \mathbf{Q}\) c. \(\mathbf{Q} \subseteq \mathbf{Z}\) d. \(\mathbf{Z}^{-} \cup \mathbf{Z}^{+}=\mathbf{Z}\) e. \(\mathbf{Z}^{-} \cap \mathbf{Z}^{+}=\emptyset\) f. \(\mathbf{Q} \cap \mathbf{R}=\mathbf{Q}\) g. \(\mathbf{Q} \cup \mathbf{Z}=\mathbf{Q}\) h. \(\mathbf{Z}^{+} \cap \mathbf{R}=\mathbf{Z}^{+}\) i. \(\mathbf{Z} \cup \mathbf{Q}=\mathbf{Z}\)

Let sets \(R, S\), and \(T\) be defined as follows: $$ \begin{aligned} &R=\\{x \in \mathbf{Z} \mid x \text { is divisible by } 2\\} \\ &S=\\{y \in \mathbf{Z} \mid y \text { is divisible by } 3\\} \\ &T=\\{z \in \mathbf{Z} \mid z \text { is divisible by } 6\\} \end{aligned} $$ a. Is \(R \subseteq T\) ? Explain. b. Is \(T \subseteq R ?\) Explain. c. Is \(T \subseteq S ?\) Explain.

Prove each statement that is true and find a counterexample for each statement that is false. Assume all sets are subsets of a universal set \(U\). For all sets \(A\) and \(B, \mathscr{P}(A) \cup \mathscr{P}(B) \subseteq \mathscr{P}(A \cup B)\).

Draw Venn diagrams to describe sets \(A, B\), and \(C\) that satisfy the given conditions. a. \(A \cap B=\emptyset, A \subseteq C, C \cap B \neq \emptyset\) b. \(A \subseteq B, C \subseteq B, A \cap C \neq \emptyset\) c. \(A \cap B \neq \emptyset, B \cap C \neq \emptyset, A \cap C=\emptyset, A \nsubseteq B, C \nsubseteq B\)

Consider the Venn diagram shown in the next column. For each of (a)-(f), copy the diagram and shade the region corresponding to the indicated set. a. \(A \cap B\) b. \(B \cup C\) C. \(A^{c}\) d. \(A-(B \cup C)\) e. \((A \cup B)^{c}\) f. \(A^{c} \cap B^{c}\)

Let \(E\) be the set of all even integers and \(O\) the set of all odd integers. Is \(\\{E, O\\}\) a partition of \(\mathbf{Z}\), the set of all integers? Explain your answer.

Use the element method for proving a set equals the empty set to prove each statement in 23-34. Assume that all sets are subsets of a universal set \(U\). For all sets \(A, B\), and \(C\), $$ (A-C) \cap(B-C) \cap(A-B)=\emptyset $$

$$ \text { If } U \text { denotes a universal set, then } U^{c}=\emptyset \text {. } $$