Regular languages are languages that can be generated by applying operations like concatenation, union, and kleene closure a finite number of times. They can be recognized by finite automata. The document provides examples of regular expressions and the regular languages they define, such as: (a) a followed by 0 or more b defines the language {a, ab, abb, abbb, ...} (b) any number of vowels followed by any number of consonants defines the language {ε, a, aou, aiou, b, abcd, ...} It also proves that the union and intersection of two regular sets is also a regular set by providing examples of regular expressions for the union and intersection of

1. Prof.Neeraj Bhargava
Abhishek Kumar
Department of Computer Science
School of Engineering & System
Sciences,
MDS, University Ajmer, Rajasthan,
India
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Regular languages are languages that can be generated from
one-element languages by applying certain standard operations a
finite number of times.
 They are the languages that can be recognized by finite
automata. These simple operations include concatenation, union
and kleen closure.
By the use of these operations regular languages can be
represented by an explicit formula.

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4. BASE REGULAR EXPRESSION
REGULAR
LANGUAGES
set of vovels ( a ∪ e ∪ i ∪ o ∪ u ) {a, e, i, o, u}
a followed by 0 or
more b
(a.b*)
{a, ab, abb, abbb,
abbbb,….}
any no. of vowels
followed by any no. of
consonants
v*.c* ( where v – vowels
and c – consonants)
{ ε , a ,aou, aiou, b,
abcd…..} where ε
represent empty string
(in case 0 vowels and
o consonants )
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Any set that represents the value of the Regular Expression is
called a Regular Set.
Property 1. The union of two regular set is regular.
Proof −
Let us take two regular expressions
RE1 = a(aa)* and RE2 = (aa)*
So, L1 = {a, aaa, aaaaa,.....} (Strings of odd length
excluding Null)
and L2 ={ ε, aa, aaaa, aaaaaa,.......} (Strings of even
length including Null)
L1 ∪ L2 = { ε, a, aa, aaa, aaaa, aaaaa, aaaaaa,.......}
(Strings of all possible lengths including Null)
RE (L1 ∪ L2) = a* (which is a regular expression itself)

6.  Property 2. The intersection of two
regular set is regular.
 Proof −
 Let us take two regular expressions
 RE1 = a(a*) and RE2 = (aa)*
 So, L1 = { a,aa, aaa, aaaa, ....} (Strings of
all possible lengths excluding Null)
 L2 = { ε, aa, aaaa, aaaaaa,.......} (Strings
of even length including Null)
 L1 ∩ L2 = { aa, aaaa, aaaaaa,.......} (Strings
of even length excluding Null)
 RE (L1 ∩ L2) = aa(aa)* which is a regular
expression itself.
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