91Ó°ÊÓ

a) For \(A=\\{1,2,3,4, \ldots, 7\\}\), how many bijective functions \(f: A \rightarrow A\) satisfy \(f(1) \neq 1 ?\) b) Answer part (a) where \(A=\left\\{x \mid x \in \mathbf{Z}^{+}, 1 \leq x \leq n\right\\}\), for some fixed \(n \in \mathbf{Z}^{+}\)

Show that if eight people are in a room, at least two of them have birthdays that occur on the same day of the week.

If \(A=\\{1,2,3\\}\), and \(B=\\{2,4,5\\}\), give examples of (a) three nonempty relations from \(A\) to \(B\); (b) three nonempty relations on \(A\).

If there are 2187 functions \(f: A \rightarrow B\) and \(|B|=3\), what is \(|A| ?\)

. Let \(S=\\{3,7,11,15,19, \ldots, 95,99,103\\}\). How many elements must we select from \(S\) to insure that there will be at least two whose sum is \(110 ?\)

a) Prove that if 151 integers are selected from \(\\{1,2,3\), \(\ldots, 300\\}\), then the selection must include two integers \(x, y\) where \(x \mid y\) or \(y \mid x\). b) Write a statement that generalizes the results of part (a) and Example 5.43.

The men's final at Wimbledon is won by the first player to win three sets of the five-set match. Let \(\mathrm{C}\) and \(\mathrm{M}\) denote the players. Draw a tree diagram to show all the ways in which the match can be decided.

a) Show that if any 14 integers are selected from the set \(S=\\{1,2,3, \ldots, 25\\}\), there are at least two whose sum is 26 . b) Write a statement that generalizes the results of part (a) and Example \(5.44 .\)

Logic chips are taken from a container, tested individually, and labeled defective or good. The testing process is continued until either two defective chips are found or five chips are tested in total. Using a tree diagram, exhibit a sample space for this process.

. Let \(A=\\{2,4,8,16,32\\}\), and consider the closed binary operation \(f: A \times A \rightarrow A\) where \(f(a, b)=\operatorname{gcd}(a, b)\). Does \(f\) have an identity element?