This is a question involving regular expressions for regular languages.
I am currently stuck trying to prove that the operand ∅ is not necessary unless the language is the empty set. That is, a regular expression is either equivalent to ∅ or is a ∅-free regular language. (A language is ∅ free if it contains no occurences of ∅)
This seems to be an induction proof, but I'm not really sure how to prove this. It seems like you're possibly trying to show that if a set is ∅ free, then doing anything involving ∅ will either result in the language ∅ or a ∅ free language.
So if r is ∅ free, then (r+∅) = r, r∅=∅ etc
I'm not really sure if this is really complete. I'm not overly comfortable with regular languages yet, so any help is much appreciated.