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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

RE_{1} = a(aa)* and RE_{2} = (aa)*

So, L_{1} = {a, aaa, aaaaa,.....} (Strings of odd length excluding Null)

and L_{2} ={ ε, aa, aaaa, aaaaaa,.......} (Strings of even length including Null)

L_{1} ∪ L_{2} = { ε, a, aa, aaa, aaaa, aaaaa, aaaaaa,.......}

(Strings of all possible lengths including Null)

RE (L_{1} ∪ L_{2}) = a* (which is a regular expression itself)

**Hence, proved.**

**Property 2.** *The intersection of two regular set is regular.*

**Proof** −

Let us take two regular expressions

RE_{1} = a(a*) and RE_{2} = (aa)*

So, L_{1} = { a,aa, aaa, aaaa, ....} (Strings of all possible lengths excluding Null)

L_{2} = { ε, aa, aaaa, aaaaaa,.......} (Strings of even length including Null)

L_{1} ∩ L_{2} = { aa, aaaa, aaaaaa,.......} (Strings of even length excluding Null)

RE (L_{1} ∩ L_{2}) = aa(aa)* which is a regular expression itself.

**Hence, proved.**

**Property 3.** *The complement of a regular set is regular.*

**Proof** −

Let us take a regular expression −

RE = (aa)*

So, L = {ε, aa, aaaa, aaaaaa, .......} (Strings of even length including Null)

Complement of **L** is all the strings that is not in **L**.

So, L’ = {a, aaa, aaaaa, .....} (Strings of odd length excluding Null)

RE (L’) = a(aa)* which is a regular expression itself.

**Hence, proved.**

**Property 4.** *The difference of two regular set is regular.*

**Proof** −

Let us take two regular expressions −

RE_{1} = a (a*) and RE_{2} = (aa)*

So, L_{1} = {a, aa, aaa, aaaa, ....} (Strings of all possible lengths excluding Null)

L_{2} = { ε, aa, aaaa, aaaaaa,.......} (Strings of even length including Null)

L_{1} – L_{2} = {a, aaa, aaaaa, aaaaaaa, ....}

(Strings of all odd lengths excluding Null)

RE (L_{1} – L_{2}) = a (aa)* which is a regular expression.

**Hence, proved.**

**Property 5.** *The reversal of a regular set is regular.*

**Proof** −

We have to prove **L ^{R}** is also regular if

Let, L = {01, 10, 11, 10}

RE (L) = 01 + 10 + 11 + 10

L^{R} = {10, 01, 11, 01}

RE (L^{R}) = 01 + 10 + 11 + 10 which is regular

**Hence, proved.**

**Property 6.** *The closure of a regular set is regular.*

**Proof** −

If L = {a, aaa, aaaaa, .......} (Strings of odd length excluding Null)

i.e., RE (L) = a (aa)*

L* = {a, aa, aaa, aaaa , aaaaa,……………} (Strings of all lengths excluding Null)

RE (L*) = a (a)*

**Hence, proved.**

**Property 7.** *The concatenation of two regular sets is regular.*

**Proof −**

Let RE_{1} = (0+1)*0 and RE_{2} = 01(0+1)*

Here, L_{1} = {0, 00, 10, 000, 010, ......} (Set of strings ending in 0)

and L_{2} = {01, 010,011,.....} (Set of strings beginning with 01)

Then, L_{1} L_{2} = {001,0010,0011,0001,00010,00011,1001,10010,.............}

Set of strings containing 001 as a substring which can be represented by an RE − (0 + 1)*001(0 + 1)*

Hence, proved.

Given R, P, L, Q as regular expressions, the following identities hold −

- ∅* = ε
- ε* = ε
- RR* = R*R
- R*R* = R*
- (R*)* = R*
- RR* = R*R
- (PQ)*P =P(QP)*
- (a+b)* = (a*b*)* = (a*+b*)* = (a+b*)* = a*(ba*)*
- R + ∅ = ∅ + R = R (The identity for union)
- R ε = ε R = R (The identity for concatenation)
- ∅ L = L ∅ = ∅ (The annihilator for concatenation)
- R + R = R (Idempotent law)
- L (M + N) = LM + LN (Left distributive law)
- (M + N) L = ML + NL (Right distributive law)
- ε + RR* = ε + R*R = R*

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