91Ó°ÊÓ

a. To say that an element is in \(A \cap(B \cup C)\) means that it is in \(\underline{(1)}\) and in \((2)\) b. To say that an element is in \((A \cap B) \cup C\) means that it is in \(\frac{(1)}{}\) or in \(\underline{(2)}\). c. To say that an element is in \(A-(B \cap C)\) means that it is in \(\frac{(1)}{}\) and not in \(\frac{(2)}{}\).

Find a counterexample to show that the statement is false. Assume all sets are subsets of a universal set \(U\). For all sets \(A, B\), and \(C,(A \cap B) \cup C=A \cap(B \cup C)\).

The following is a proof that for all sets \(A, B\), and \(C\), if \(A \subseteq B\) and \(B \subseteq C\), then \(A \subseteq C\). Fill in the blanks. Proof: Suppose \(A, B\), and \(C\) are sets and \(A \subseteq B\) and \(B \subseteq C\). To show that \(A \subseteq C\), we must show that every element in \(\frac{(1)}{}\) is in \((2)\). But given any element in \(A\), that element is in \(\underline{(3)}\) (because \(A \subseteq B\) ), and so that element is also in \(\stackrel{(4)}{\text { (because }}(5)\) ). Hence \(A \subseteq C\).

$$ \text { Prove that for all sets } A \text { and } B, B-A=B \cap A^{c} \text {. } $$

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 \(A, B\), and \(C, A-(B-C)=(A-B)-C\).

Prove that for all sets \(A\) and \(B,(A \cap B)^{c}=A^{c} \cup B^{c}\). Use an element argument to prove each statement in 8-17. Assume that all sets are subsets of a universal set \(U\).

a. Is \(3 \in\\{1,2,3\\}\) ? b. Is \(1 \subseteq\\{1\\}\) ? c. Is \(\\{2\\} \in\\{1,2\\}\) ? d. Is \(\\{3\\} \in\\{1,\\{2\\},\\{3\\}\\}\) ? e. Is \(1 \in\\{1\\}\) ? f. Is \(\\{2\\} \subseteq\\{1,\\{2\\},\\{3\\}\\}\) ? g. Is \(\\{1\\} \subseteq\\{1,2\\} ?\) h. Is \(1 \in\\{\\{1\\}, 2\\}\) ? i. Is \(\\{1\\} \subseteq\\{1,\\{2\\}\\}\) ? j. Is \(\\{1\\} \subseteq\\{1\\}\) ?

For all sets \(A, B\), and \(C\), $$ (A-B) \cap(C-B)=(A \cap C)-B $$

Let the universal set be the set \(\mathbf{R}\) of all real numbers and let \(A=\\{x \in \mathbf{R} \mid 0

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.