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Chapter 5: Problem 36
Find the reversal of the following bit strings. a) 0101 b) 11011 c) 100010010111
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Let \(S\) be the subset of the set of ordered pairs of integers defined recursively by Basis step: \((0,0) \in S .\) Recursive step: If \((a, b) \in S,\) then \((a+2, b+3) \in S\) and \((a+3, b+2) \in S\) a) List the elements of \(S\) produced by the first five appli- cations of the recursive definition. b) Use strong induction on the number of applications of the recursive step of the definition to show that \(5 | a+b\) when \((a, b) \in S .\) c) Use structural induction to show that \(5 | a+b\) when \((a, b) \in S .\)
Show that \(\left[\left(p_{1} \rightarrow p_{2}\right) \wedge\left(p_{2} \rightarrow p_{3}\right) \wedge \cdots \wedge\left(p_{n-1} \rightarrow p_{n}\right)\right]\) \(\quad \rightarrow\left[\left(p_{1} \wedge p_{2} \wedge \cdots \wedge p_{n-1}\right) \rightarrow p_{n}\right]\) is a tautology whenever \(p_{1}, p_{2}, \ldots, p_{n}\) are propositions, where \(n \geq 2\)
Show that if \(I_{1}, I_{2}, \ldots, I_{n}\) is a collection of open intervals
on the real number line, \(n \geq 2,\) and every pair of these intervals has a
nonempty intersection, that is, \(I_{i} \cap I_{j} \neq \emptyset\) whenever \(1
\leq i \leq n\) and \(1 \leq j \leq n,\) then the intersection of all these sets
is nonempty, that is, \(I_{1} \cap I_{2} \cap \cdots \cap I_{n} \neq \emptyset\)
. (Recall that an open interval is the set of real numbers \(x\) with \(a< x
Deal with values of iterated functions. Suppose that \(f(n)\) is a function from the set of real numbers, or positive real numbers, or some other set of real numbers, to the set of real numbers such that \(f(n)\) is monotonically increasing [that is, \(f(n)< f(m)\) when \(n< m )\) and \(f(n)< n\) for all \(n\) in the domain of \(f . ]\) The function \(f^{(k)}(n)\) is defined recursively by $$f^{(k)}(n)=\left\\{\begin{array}{ll}{n} & {\text { if } k=0} \\\ {f\left(f^{(k-1)}(n)\right)} & {\text { if } k>0}\end{array}\right.$$ Furthermore, let \(c\) be a positive real number. The iterated function \(f_{c}^{*}\) is the number of iterations of \(f\) required to reduce its argument to \(c\) or less, so \(f_{c}^{*}(n)\) is the smallest nonnegative integer \(k\) such that \(f^{k}(n) \leq c\). Let \(f(n)=n / 2 .\) Find a formula for \(f^{(k)}(n) .\) What is the value of \(f_{1}^{*}(n)\) when \(n\) is a positive integer?
Show that each of these proposed recursive definitions of a function on the set of positive integers does not produce a well-defined function. a) \(F(n)=1+F(\lfloor n / 2\rfloor)\) for \(n \geq 1\) and \(F(1)=1\) b) \(F(n)=1+F(n-3)\) for \(n \geq 2, \quad F(1)=2, \quad\) and \(F(2)=3\) c) \(F(n)=1+F(n / 2)\) for \(n \geq 2, F(1)=1,\) and \(F(2)=2\) d) \(F(n)=1+F(n / 2)\) if \(n\) is even and \(n \geq 2, F(n)=1-\) \(F(n-1)\) if \(n\) is odd, and \(F(1)=1\) e) \(F(n)=1+F(n / 2)\) if \(n\) is even and \(n \geq 2, F(n)=$$F(3 n-1)\) if \(n\) is odd and \(n \geq 3,\) and \(F(1)=1\)
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