Answer
We try to prove it with the help of proving by contradiction technique and get these answers.
$A$nita is $knave$.
$B$oris is $knave$
$C$armen is $knave$
Work Step by Step
Provided Data:
$A$nita says" I am a $knave$ and $B$oris is a $knight$"
$B$oris says" Exactly one of us is a $knight$"
Specifications:
A $knight$ always tells the truth.
A $knave$ always lies.
Let us assume that $B$oris is a $knight$, then $B$oris tells the truth and thus $B$oris is a $knight$, while the others are $knaves$. However, we then note that $A$nita’s statement is also true, but $A$nita is a $knave$ thus the statement cannot be true. We thus obtained of contradiction and thus $B$oris cannot be a $knight$.
$B$oris is not a $knight$, thus $B$oris is a $knave$.
$B$oris is not a $knight$, thus $A$nita’s statement is false and thus $A$nita has to be a $knave$ as well.
$B$oris is $knave$, $B$oris’s statement is false and thus there cannot be exactly one $knight$. If $C$armen is a $knight$, then there would be exactly one $knight$. Thus it is impossible that $C$armen is a $Knight$ and thus $C$armen is a $knave$ as well.
