Are the languages of the following regular expressions the same?

(a|b)*

(b|a)*

Question

Are the languages of the following regular expressions the same?

(a|b)*

(b|a)*

BlackTom AI · Accepted Answer

Question restatement: Are the languages of the following regular expressions the same?

(a|b)*

(b|a)*

Options: True, False

Option 1: True — Why this might be considered correct. The set (a|b) denotes a choice between symbol a or symbol b. When you apply Kleene star to (a|b), you obtain all strings that are sequences of a and b in any order, including the empty string. Since the union (a|b) contains exactly the two symbols in the alphabet, repeating them in any order with any length produces every possible string over the alphabet {a,b} (including the empty string). The same reasoning applies to (b|a)* because the union (b|a) also consists of the same two symbols {a,b} and the star allows any finite concatenation of these symbols. Therefore, both expressions generate the same language: all strings over {a,b}.

Option 2: False — Why this would be incorrect if considered. It might seem that the order within the union could affect the language, but for a finite alphabet with the same symbols, the order of alternatives in a union does not constrain the resulting set after applying Kleene star. The star allows any number of concatenations, so the distinction between (a|b) versus (b|a) collapses to the same collection of strings, namely all possible combinations of a and b, including the empty string.

Summary of reasoning for both options. The correctness hinges on understanding that the star over a union of symbols yields all strings formed from those symbols, and the set {a,b} is identical regardless of the order of a and b inside the union. Since both regular expressions effectively specify the same alphabet and the same closure under repetition, they generate the same language. If you imagined a different alphabet or a different operator, the conclusion could change, but in this case the two expressions align in their language.

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Are the languages of the following regular expressions the same? (a|b)* (b|a)*

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