# Closure Properties of regular expressions [Union Intersection Concatenation]

## Closure Properties of Regular Expressions

- Union
- Intersection
- concatenation
- Kleene closure
- Complement

## Union Of Regular Expression With Examples

If RE1 and RE2 are regular, then RE1 union RE2 is also a regular expression. This can be written as;

- RE1 | RE2
- RE1 U RE2
- RE1 + RE2

## Examples of Union Operation Of Regular Expression

**Example 1**

**RE1**= a(b)*

**RE2** = b(a)*

**RE1 + RE2**= a(b)* + b(a)*

**Result:** Regular expressions are closed under union operation.

**Example 2**

**Regular expression 1:** a

**Regular expression 2:** b

**Regular expression 1 + Regular expression 2** = a+b

## More examples of Union as a closure property Of Regular Expression

**Regular expression 1:** a(a)

**Regular expression 2:** b(a+b)

**Regular expression 1 + Regular expression 2** = a(a)+ b(a+b)

**Regular expression 1:** b(aa+b)

**Regular expression 2:** b(aa+b)

**Regular expression 1 + Regular expression 2** = b(aa+b) + b(aa+b)

**Regular expression 1:** a(aa+bb)*

**Regular expression 2:** b(aa)

**Regular expression 1 + Regular expression 2** = a(aa+bb)*+ b(aa)

**I****ntersection** Of Regular Expression With Examples

If RE1 and RE2 are regular, then

RE1 ∩ RE2

is also a regular expression.

## Examples of Intersection Operation Of Regular Expression

## Example 1:

RE1 = b(b*)

RE2 = (bb)*

RE1 ∩ RE2= bb(bb)*

**Explanation:**

L_{1} = { b,bb, bbb, bbbb, ….} (Strings of all possible lengths of b without Null)

L_{2} = { ε, bb, bbbb, bbbbbb,…….} (Strings of even length of b including Null)

L_{1} ∩ L_{2} = { bb, bbbb, bbbbbb,…….} (Strings of even length of b excluding Null)

# Concatenation Of Regular Expression With Examples

If RE1 and RE2 are regular, then

RE1.RE2

is also a regular expression.

## Examples of Concatenation Operation Of Regular Expression

## Example 1:

**Regular Expression 1** = ( 00 + 10 )

strings are = {00, 10 only}

**Regular Expression 2** = ( 0 + 1 )

strings are = {0, 1 only}

**Regular Expression 1 ◦Regular Expression 2** = ( 00 + 10 )( 0 + 1 )

strings are = {000, 001, 100, 101}

## Example 2:

**Regular Expression 1 **= (0+1)*0

set of strings ending in 0

**Regular Expression 2 **= 01(0+1)*

set of strings beginning with 01

**Regular Expression 1** ◦ **Regular Expression 2** = 01(0+1)* (0+1)*0 which can be written in more optimized way as;

01(0+1)*0