Concatenation of language to itself zero times I was solving this question:


  Which of the following statement(s) is/are false? 
  
  
  $L^0=\{\epsilon\}$
  $|L^0|=0$
  


The answer given was None. That is, none of these statements are false and both are true. However, I feel both are wrong. I feel

$L^1=L$  and
$L^0=\phi$

Also 

$|L^0|=|\phi|=0$

Am I wrong with this?
$L^0$ is defined to be $\{\varepsilon\}$. (Since a string formed by concatenating zero strings together is the empty string).

This gives the answer to the first question - which is correct.

Now, $\{\varepsilon\}$ contains exactly one string - which is the empty string. Therefore: $|\{\varepsilon\}|=1$. (The given answer is wrong).

Note that $|\{\varepsilon\}|=1\ne0=|\varepsilon|$. Also, note that $\{\varepsilon\} \ne \emptyset$.