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Regular sets and their properties in Theory of Computation

Here, we are going to learn about the regular sets and their properties in theory of computation.
Submitted by Mahak Jain, on November 14, 2018

Any set that denotes the value of the Regular Expression is called a "Regular Set".

Regular sets have various properties:

Property 1) The union of two regular sets is also a regular set

Proof:

    Let there be two regular expressions
    RE1 = 0(00)* and RE2 = (00)*
    So, L1 = {0, 000, 00000,..} (odd length strings without Null)
    and L2 ={ ε, 00, 0000, 000000,.......} (even length stringswith Null)
    L1 ∪ L2 = { ε, 0, 00, 000, 0000, 00000, 000000,.......}
    (all possible lengths stringswith Null)
    RE (L1 ∪ L2) = 0* (this is also a regular expression or a regular set)

Hence, proved.

Property 2) The intersection of two regular set is also a regular set.

Proof:

    Let there be two regular expressions
    RE1 = 0(0*) and RE2 = (00)*
    So, L1 = { 0,00, 000, 0000, ....} (odd length strings without Null)
    L2 = { ε, 00, 0000, 000000,.......} (even length strings with Null)
    L1 ∩ L2 = { 00, 0000, 000000,.......} (even length stringswithout Null)
    RE (L1 ∩ L2) = 00(00)*(this is a regular set or regular expression also).

Hence, proved.

Property 3) The complement of a regular set is also regular set.

Proof:

    Let there be a regular expression −
    RE = (00)*
    So, L = {ε, 00, 0000, 000000, .......} (even length stringswith Null)
    Complement of L is set of all the strings that is not included in L.
    So, L' = {0, 000, 00000, .....} (odd length stringswithout Null)
    RE (L') = 0(00)*this denotes a regular set or regular expression in itself.

Hence, proved.

Property 4) The difference of two regular set is also a regular set.

Proof:

    Let there be two regular expressions −
    RE1 = 0 (0*) and RE2 = (00)*
    So, L1 = {0, 00, 000, 0000, ....} (all possible lengths stringswithout Null)
    L2 = { ε, 00, 0000, 000000,.......} (even length strings with Null)
    L1 – L2 = {0, 000, 00000, 0000000, ....}
    (all odd lengths strings without Null)
    RE (L1 – L2) = 0 (00)*(this is also a regular set or a 
    regular expression in itself)

Hence, proved.

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Property 5) The reversal of a regular set is also a regular set.

Proof:

    If L is a regular set:
    Let, L = {ab, ba, bb, ba}
    RE (L) = b1 + 1b +bb + ba
    LR = {ba, ab, bb, ab}
    RE (LR) = ab + ba + bb + bathis is the reverse and also a regular 
    set or a regular expression in itself 

Hence, proved.

Property 6) The closure of a regular set or an expression is also a regular set.

Proof:

    If L = {0, 000, 00000, .......} (all odd length strings without Null)
    i.e., RE (L) = 0 (00)*
    L* = {0, 00, 000, 0000 ,00000,……………} ( all lengths strings without Null)
    RE (L*) = 0 (0)*(this is also a regular set or a regular expression) 

Hence, proved.

Property 7) The concatenation of two regular sets is also a regular set.

Proof:

    Let RE1 = (a+1)*a and RE2 = a1(a+1)*
    Here, L1 = {a, aa, 1a, aaa, a1a, ......} (Set of strings ending in a)
    and L2 = {a1, a1a,a11,.....} (Set of strings beginning with a1)
    Then, L1 L2 = {aa1,aa1a,aa11,aaa1,aa1a,aaa11,1aa1,1aa1a,.............}
    Set of strings containing aa1 as a substring which 
    can be denoted by an RE − (a + 1)*aa1(a + 1)*

Hence, proved.

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