Question

Let L ⊆ Σ^{*} be a regular language. Suppose a ∈ Σ and
define L\a = {x : ax ∈ L }. Show that L\a is regular.

Answer #1

Let Σ = {a}, and let L be the language
L={an :nisamultipleof3butnisNOTamultipleof5}.
Is L a regular language? HINT: Maybe instead of an explicit DFA
or regular expression, you can find another argument.

Given a language L, etc.
Show that the language L is a regular language.
To show that the language L is a regular language - find/design a
dfa that recognizes the language L.
Given a regular expression r, etc.
What is the language L, L = L(r)?
L(r) is the set of all strings etc.

For Automata class:
Let L be a regular language over the binary alphabet. Consider
the following language over the same alphabet: L' = {w | |w| = |u|
for some u ∈ L}. Prove that L' is regular.

5 A Non-Regular language
Prove that the language}L={www∣w∈{0,1}∗} is not regular.

Let Σ = {0,1}. Prove that the language { w | w contains the
substring 01001 } is regular by providing a finite automaton to
recognize the language. Include a state diagram, formal
description, and informal justification for the correctness of your
automaton.

Let L1 be the language of the Regular Expression 1(1
+ 0)*.
Let L2 be the language of the Regular Expression 11*
0.
Let L3 be the language of the Regular Expression 1*
0.
Which of the following statements are true?
L2 L1
L2 L3
L1 L2
L3 L2

Prove by induction on n that if L is a language and R is a
regular expression such that L = L(R) then there exists a regular
expression Rn such that L(Rn) = L n. Be sure to use the fact that
if R1 and R2 are regular expressions then L(R1R2) = L(R1) ·
L(R2).

Suppose A and B are regular language. Prove that AB is regular.

Let swap_every_two be an operation on languages that is defined
as follows:
swap_every_two(L) = {a2a1a4a3 . . . a2na2n−1 | a1a2a3a4 . . .
a2n−1a2n ∈ L where a1, . . . , a2n ∈ Σ} In this definition, Σ is
the alphabet for the language L.
1. What languages result from applying swap every two to the
following languages:
(a) {1 n | n ≥ 0}, where the alphabet is {1}.
(b) {(01)n | n ≥ 0}, where the...

Let
X be a random variable with mean μ and variance σ^2. Define
Y=(X-μ)/σ. What is the variance of Y?

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