Is a single string enough to prove regular expressions inequivalent? Which of the following regular expressions generate a language that is different from the rest?    
  
  
  (a+b)$^*$a(a+b)$^*$(a+b)$^*$  
  b$^*$ab$^*$a(a+b)$^*$  
  (a+b)$^*$ab$^*$ab$^*$  
  b$^*$ a(a+b)$^*$ab$^*$  
  


RE 1 generates a language that contains the string 'a' which no other language among 2,3,4 contains, so can I say this language is different from all others?  Does it suffice to show that a single string is not present in the language and hence its different from the rest?
Does it suffice to show that a single string is not present in the language and hence its different from the rest? 


Yes. It's actually a very neat proof.

Formally speaking, you have found $w \in \{a,b\}^*$ with $w \in L_1$ but $w \not\in L_2,L_3,L_4$. That is sufficient to show that $L_1 \neq L_2$, $L_1 \neq L_3$ and $L_1 \neq L_4$.

You have not shown that $L_2 = L_3 = L_4$. The phrasing of the question seems to suggest as much (MC questions are boring!) but you may want to prove that as an additional exercise.


   Hint: Translate into NFA, determinize, minimize, check for isomorphy.