# Condition in Arden's rule

According to Arden's rule, the language equation $$X= AX\cup B$$, with unknown $$X$$, has the solution $$X=A^*B$$, provided $$A$$ does not contain the empty string.

My question: what is the problem with the empty string here? Can you illustrate this with a concrete example (which lacks in Wikipedia on this topic).

• – Hendrik Jan Dec 14 '19 at 3:39

Consider, for instance, $$A = \{ \varepsilon \}$$. Then $$X = AX$$, so $$X = AX \cup B$$ holds for any $$X$$ with $$B \subseteq X$$.