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Let $\mathrm{L}$ be a regular language. Show that the following languages are regular. a) The set $\mathrm{P}=\{u \mid u v \in \mathrm{L}\}$ of prefixes of $\mathrm{L}$ b) The set $\mathrm{S}=\{v \mid u v \in \mathrm{L}\}$ of suffixes of $\mathrm{L}$ c) The set $\mathbf{L}^R=\left\{w^R \mid w \in \mathrm{L}\right\}$ of reversals of $\mathrm{L}$ d) The set $\mathrm{E}=\{u v \mid v \in \mathrm{L}\}$ of strings that have a suffix in $\mathrm{L}$

    Let $\mathrm{L}$ be a regular language. Show that the following languages are regular.
a) The set $\mathrm{P}=\{u \mid u v \in \mathrm{L}\}$ of prefixes of $\mathrm{L}$
b) The set $\mathrm{S}=\{v \mid u v \in \mathrm{L}\}$ of suffixes of $\mathrm{L}$
c) The set $\mathbf{L}^R=\left\{w^R \mid w \in \mathrm{L}\right\}$ of reversals of $\mathrm{L}$
d) The set $\mathrm{E}=\{u v \mid v \in \mathrm{L}\}$ of strings that have a suffix in $\mathrm{L}$

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Languages and Machines: An Introduction to the Theory of Computer Science

Languages and Machines: An Introduction to the Theory of Computer Science

Thomas A. Sudkamp 2nd Edition

Chapter 7, Problem 8

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Step 1

- Since $\mathrm{L}$ is a regular language, there exists a deterministic finite automaton (DFA) $M = (Q, \Sigma, \delta, q_0, F)$ that recognizes $\mathrm{L}$. - Construct a new DFA $M' = (Q, \Sigma, \delta, q_0, F')$ where $F' = \{q \in Q \mid \exists v \in Show more…

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Let $\mathrm{L}$ be a regular language. Show that the following languages are regular. a) The set $\mathrm{P}=\{u \mid u v \in \mathrm{L}\}$ of prefixes of $\mathrm{L}$ b) The set $\mathrm{S}=\{v \mid u v \in \mathrm{L}\}$ of suffixes of $\mathrm{L}$ c) The set $\mathbf{L}^R=\left\{w^R \mid w \in \mathrm{L}\right\}$ of reversals of $\mathrm{L}$ d) The set $\mathrm{E}=\{u v \mid v \in \mathrm{L}\}$ of strings that have a suffix in $\mathrm{L}$

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