Necessary and Sufficient Conditions for x and A

Question

Having $A^* - \{x\} = A^+$ while $A$ being a language over $\{a,b\}$ and $x∈\{a,b\}^*$

This is only true $iff$ $x = ε$

So I did a proof:

$A^* -\{ ε \} = A^+\\ A^*-\{ε\}+\{ε\}=A^++\{ε\}\\ A^*=A^*\cdot A + \{ε\}\\ A^*=A^*$

Is this enough to comply with the "necessary and sufficient" conditions? I'm asking because I was told it wasn't. The original question asked:

'Find necessary and sufficient conditions in terms of x and A for the equation'

Thanks

Answer 1

There are two cases here to consider:

1. If $\def\e{\varepsilon} \e\in A$, we have $A^*=A^+$, and then $x$ has to be [can be] any word $\notin A$, so that $A^*-\{x\}\ =\ A^*\ =A^+$.
2. If $\e\notin A$, indeed we have $A^*=\{\e\}+A^+$, and then indeed only $x=\e\ $ works.