91Ó°ÊÓ

How many permutations of the 26 different letters of the alphabet contain (a) either the pattern "OUT" or the pattern "DIG"? (b) neither the pattern "MAN" nor the pattem "ANT"?

Which of the following sets are equal? a) \(\\{1,2,3\\}\) b) \(\\{3,2,1,3\\}\) c) \(\\{3,1,2,3\\}\) d) \(\\{1,2,2,3\\}\)

For \(q_{\ell}=\\{1,2,3, \ldots, 9,10\\}\) let \(A=\\{1,2,3,4,5\\}, B=\\{1,2,4,8\\}, C=\\{1,2,3,5,7\\}\), and \(D=\) \(\\{2,4,6,8\\}\). Determine each of the following: a) \((A \cup B) \cap C\) b) \(A \cup(B \cap C)\) c) \(\bar{C} \cup \bar{D}\) d) \(C \cap D\) e) \((A \cup B)-C\) f) \(A \cup(B-C)\) g) \((B-C)-D\) h) \(B-(C-D)\) i) \((A \cup B)-(C \cap D)\)

Let \(A, B, C \subseteq ?\). Prove that \((A-B) \subseteq C\) if and only if \((A-C) \subseteq B\).

If \(A, B \subseteq ?\), prove that \(A \subseteq B\) if and only if \([\forall C \subseteq \mathcal{U}(C \subseteq A \Rightarrow C \subseteq B)]\).

If \(A=[0,3], B=[2,7)\), with \(\Psi=\mathbf{R}\), determine each of the following: a) \(A \cap B\) b) \(A \cup B\) c) \(\bar{A}\) d) \(A \triangle B\) e) \(A-B\) f) \(B-A\)

Let \(A=\\{1,\\{1\\},\\{2\\}\\}\). Which of the following statements are true? a) \(1 \in A\) b) \(\\{1\\} \in A\) c) \(\\{1\\} \subseteq A\) (d) \(\\{\\{1\\} \backslash A\) e) \(\\{2\\} \in A\) f) \(\\{2\\} \subseteq A\) g) \(\\{\\{2\\}\\} \subseteq A\) h) \(\\{\\{2\\}\\} \subset A\)

Let \(A, B, C \subseteq 9\). Prove or disprove (with a counterexample) each of the following: a) \(A-C=B-C \Rightarrow A=B\) b) \([(A \cap C=B \cap C) \wedge(A-C=B-C)] \Rightarrow A=B\) c) \([(A \cup C=B \cup C) \wedge(A-C=B-C)] \Rightarrow A=B\) d) \(\mathscr{( A - B )}=\mathscr{P}(A)-\mathscr{P}(B)\)

a) For positive integers \(m, n, r\), with \(r \leq \min \\{m, n\\}\), show that $$ \begin{gathered} \left(\begin{array}{c} m+n \\ r \end{array}\right)=\left(\begin{array}{c} m \\ 0 \end{array}\right)\left(\begin{array}{l} n \\ r \end{array}\right)+\left(\begin{array}{c} m \\ 1 \end{array}\right)\left(\begin{array}{c} n \\ r-1 \end{array}\right)+\left(\begin{array}{c} m \\ 2 \end{array}\right)\left(\begin{array}{c} n \\ r-2 \end{array}\right) \\ +\cdots+\left(\begin{array}{c} m \\ r \end{array}\right)\left(\begin{array}{l} n \\ 0 \end{array}\right)=\sum_{k=0}^{r}\left(\begin{array}{c} m \\ k \end{array}\right)\left(\begin{array}{c} n \\ r-k \end{array}\right) . \end{gathered} $$ b) For \(n\) a positive integer, show that $$ \left(\begin{array}{l} 2 n \\ n \end{array}\right)=\sum_{k=0}^{n}\left(\begin{array}{l} n \\ k \end{array}\right)^{2} . $$

a) Determine the sets \(A, B\) where \(A-B=\\{1,3,7,11\\}, B-A=\\{2,6,8\\}\), and \(A \cap B=\) \(\\{4,9\\}\). b) Determine the sets \(C, D\) where \(C-D=\\{1,2,4\\}, D-C=\\{7,8\\}\), and \(C \cup D=\) \(\\{1,2,4,5,7,8,9\\}\).