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**Closure Properties**

- A set is
**closed**under an operation if applying that operation to any members of the set always yields a member of the set. the positive integers are closed under addition and multiplication, but not division.*For example,*- Various
*Closure Properties*are as follows −**Union****Intersection****Concatenation****Kleene closure****Complement**

**Regular Sets**

- Any set that represents the value of the
is called a*Regular Expression***Regular Set.** - A language or set is called
if it is accepted by a*Regular**Finite-State Automaton.* - In an automata theory, there are different
for*Closure Properties**Regular Languages or Sets*. - Let A and B be languages (remember they are sets).
Concatenation operation can be defined for these two regular sets as follows:

**AB = {wv : w ∈ A and v ∈ B}**

**Theorem -**

*The class of regular languages or sets is closed under union, intersection, complementation, concatenation, and kleene closure.*

Here, we see the *Concatenation Property of two Regular Sets.*

**The Regular Sets are Closed under Concatenation**

**Proof -**

Prove for arbitrary Regular Sets or Languages $L1$ and $L2$ that $L1.L2$ is a regular language.

– Let $E1$ and $E2$ be REGEX accepting $L1$ and $L2$.

– **REGEX Construction:**

We claim the REGEX

$$E = E1.E2$$

accepts $L1.L2,$ i.e. $L(E1.E2) = L(E1).L(E2)$

– **Proof of correctness:** *Trivial by definition of Regular Expressions.*

– $L1.L2$ is regular since there is a REGEX $E1.E2$ accepting this language.

**Examples -**

**1]**

E1 = (0+1)*0

E2 = 01(0+1)*

Where,

L1 = {0, 00, 10, 000, 010, ......} (Set of strings ending in 0)

and L2 = {01, 010,011,.....} (Set of strings beginning with 01)

Therefore,

L1.L2 = {001,0010,0011,0001,00010,00011,1001,10010,.............}

Set of strings containing 001 as a substring which can be represented by an

$E − (0 + 1)^*001(0 + 1)^*$

Hence, proved.

**2]**

L1 = {an | n > 0} and L2 = {bn | n > O}

**L3 = L1.L2 = {am . bn | m > 0 and n > O}**

Hence, proved.

*A Regular Set has an FA or an RE. Regular Sets or Languages are closed under Union,
Concatenation, Intersection, Complementation, and Kleene Closure.*