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Question about regular expressions. I'm trying to prove that

A^{}=A^{*}

if $B ...

Feb 4, 2024

Question about regular expressions. I'm trying to prove that

A^{*} B^{*}=A^{*}

B \subset A

Solution by Steps

step 1

Consider the definition of the Kleene star operator for a set

S

, which is

S^*

. This represents the set of all strings that can be made by concatenating zero or more elements from

S

, including the empty string

step 2

Given that

B \subset A

, every element of

B

is also an element of

A

. Therefore, any string formed by elements of

B

can also be formed by elements of

A

step 3

The set

A^* B^*

represents all strings that can be formed by concatenating zero or more elements from

A

followed by zero or more elements from

B

. Since

B \subset A

, this is equivalent to just concatenating elements from

A

step 4

Therefore,

A^* B^*

is equivalent to

A^*

because concatenating additional elements from

B

does not introduce any new strings that are not already in

A^*

Answer

A^* B^* = A^*

B \subset A

Key Concept

Kleene Star and Subset Relations in Regular Expressions

Explanation

The key concept is understanding how the Kleene star operator works and how the subset relation affects the concatenation of strings from two sets in regular expressions. If all elements of

B

are also in

A

, then

A^* B^*

does not produce any strings that are not already produced by

A^*

alone.