Does left factoring CFG make it unambiguous? I came across following problem:


  If the CFG is left factored then it must be Unambiguous and Not left Recursive.
  TRUE/FALSE?


I have many thoughts about this. But I feel they are somewhat contradicting and hence form a bigger doubt. May be I lack some understanding.

Thought 1

Ambiguity has nothing to do with determinism (even if grammar does not have productions with same prefix of their right side, they still can have a string with two parse trees). So CFG is still ambiguous after left refactoring. Hence, the sentence should be false. 

Thought 2

Left factoring is used to eliminate common prefixes of right hand side of productions in the grammar. Doesnt that make grammar deterministic? Deterministic grammar derive deterministic context free languages. So isnt turning grammar deterministic makes it unambiguous? If yes, given statement should be true. 

Thought 3 

I have completely ignored left recursion in thought 1 and 2. Is it ok?

Doubt: I feel thought 1 and  thought 2 are contradicting. And I am not able to figure it out which one is correct. Which of the above thoughts are incorrect and why? Am I missing any more subtleties?
The answer is no. Some context-free languages are inherently ambiguous: every context-free grammar defining a language of this kind will be ambiguous. An example is

$$\{a^ib^jc^k \mid i = j \text{ or } j = k\}$$