Let  A be the set of all determinants of order 3 with entries 0 or 1 only, B is the subset of  A consisting of all determinants with value 1, and C is the subset consisting of all determinants with value -1. Then if  $n\left(B\right)$ and  $n\left(C\right)$ denote the number of elements in B and C, respectively, we have

Since,  C cannot be the empty set, Hence
 $-1=\left|\begin{array}{ccc}0& 1& 0\\ 1& 0& 0\\ 0& 0& 1\end{array}\right|\in C$
We also have
 $\left|\begin{array}{ccc}1& 0& 1\\ 1& 1& 0\\ 0& 1& 1\end{array}\right|=2$
So  $A\ne B\cup C$. In general, the determinant
 $\Delta =\left|\begin{array}{ccc}{a}_{11}& a_{12}& a_{13}\\ a_{21}& a_{22}& a_{23}\\ a_{31}& a_{32}& a_{33}\end{array}\right|$
 $\Delta =a_{11}{a}_{22}{a}_{33}+a_{12}{a}_{23}{a}_{31}+a_{13}{a}_{21}{a}_{32}-a_{11}{a}_{23}{a}_{32}-a_{12}{a}_{21}{a}_{33}-a_{13}{a}_{31}{a}_{22}$ 
With the $a\text{'}s$ being 0 or 1, equals 1 only if  $a_{11}{a}_{22}{a}_{33}=1$ and the remaining terms are zero; if  $a_{12}{a}_{23}{a}_{31}=1$ and the remaining terms are zero; or if  $a_{13}{a}_{21}{a}_{32}=1$ and the remaining terms are zero. Since there are three similar relations for determinants that equal  -1, we must have $n\left(B\right)=n\left(C\right)$