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 their corresponding regular languages, including (a.b*) representing strings with an a followed by zero or more bs, and v*.c* representing strings with any number of vowels followed by any number of consonants. It also outlines two properties of regular languages: 1) the union of two regular sets is regular, and 2) the intersection of two regular sets is regular, providing proofs using examples.

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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