Are the languages of the following regular expressions the same?

a a*

a* a

Question

Are the languages of the following regular expressions the same?

a a*

a* a

BlackTom AI · Accepted Answer

The question asks whether the languages of the two regular expressions are the same: 'a a*' and 'a* a'.
Option 1: True. This would claim that 'a a*' and 'a* a' denote the same set of strings. To evaluate, note that 'a a*' means an 'a' followed by zero or more 'a's, which yields strings of one or more 'a's, i.e., a+. 
Option 2: False. This would claim the two expressions do not generate the same language. However, 'a* a' denotes zero or more 'a's followed by an 'a', which also yields strings of one or more 'a's, i.e., a+.
Delving deeper, both expressions require at least one 'a' in the string, and any string consisting solely of 'a's with length at least 1 can be formed by both patterns: for 'a a*', take the initial 'a' plus any number of trailing 'a's from the a*, and for 'a* a', take any number (including zero) of initial 'a's from a*, followed by the final 'a'. In either case, the result is exactly the set {a^n | n ≥ 1}.
Thus, the language produced by 'a a*' is the same as the language produced by 'a* a'. Therefore the statement claiming the languages are the same is correct, and the alternative option is incorrect.
If one were to consider potential ambiguity due to spacing in the expression notation, the standard interpretation without spaces treated as explicit concatenation aligns with the above conclusions: both expressions describe strings of one or more 'a' characters.

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Are the languages of the following regular expressions the same? a a* a* a

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