I thought that, for every CFG, there is an equivalent grammar in CNF. I was told this is false. Can someone explain to me why this is false and possibly provide a counter example?
For every context-free grammar, is there an equivalent grammar in Chomsky normal form?
context free;formal grammars;normal forms
Depends on how strict you interpret equivalent. A CFG in ChNF cannot generate the empty string. If you want to have a more precise statement:For every CFG $G$ there is a grammar $G'$ in ChNF such that   $L(G) = L(G') - \{\varepsilon\}$.So now we have two choices.Either to accept the loss of the empty string, which in practise is not a big deal. Or we have to adapt the definition to make room for generating the empty string in a special way. This usually means we allow a production $S\to \varepsilon$ where $S$ is the axiom, provided $S$ does not occur as right hand side of productions.This is a matter of taste: there is no best definition. Personally I think having a special production for $\varepsilon$ is not elegant, and prefer a notion of equivalence upto the empty string.  