5. Let $ S $ be the set of all finite strings of $ a $'s and $ b $'s. Define $ f : S \rightarrow \mathbb{Z} $ as follows: For each string $ s $ in $ S $,\[f(s) = \begin{cases} \text{the number of $ a $'s in $ s $ if $ s $ begins with an $ a $} \\\text{the number of $ b $'s in $ s $ if $ s $ begins with a $ b $} \\0 \text{ if $ s = \varepsilon $, that is, if $ s $ is the empty word}\end{cases}\]Find the following:(a) $ f(aba) = $(b) $ f(bbab) = $(c) What is the range of $ f $? Explain.

S is the set of all finite strings of a's and b's f:S→ℤ is defined as follows: For each string s in…

5. Let S be the set of all finite strings of a's and b's. . Define f : S → Z as follows: For each string s in S, the number of a's in s if s begins with an a f(s) = the number of b's in s if s begins with a b 0 if s = €, that is, if s is the empty word Find the following: (a) f(aba) = (b) f(bbab) = (c) What is the range of f? Explain.

5. Let S be the set of all finite strings of a's and b's. . Define f : S → Z as follows: For each string s in S, the number of a's in s if s begins with an a f(s) = the number of b's in s if s begins with a b 0 if s = €, that is, if s is the empty word Find the following: (a) f(aba) = (b) f(bbab) = (c) What is the range of f? Explain.

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Question

5. Let $ S $ be the set of all finite strings of $ a $'s and $ b $'s. Define $ f : S \rightarrow \mathbb{Z} $ as follows: For each string $ s $ in $ S $,

\[
f(s) =
\begin{cases}
\text{the number of $ a $'s in $ s $ if $ s $ begins with an $ a $} \\
\text{the number of $ b $'s in $ s $ if $ s $ begins with a $ b $} \\
0 \text{ if $ s = \varepsilon $, that is, if $ s $ is the empty word}
\end{cases}
\]

Find the following:

(a) $ f(aba) = $

(b) $ f(bbab) = $

(c) What is the range of $ f $? Explain.

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