I have two process that l will perform.
In the first process l have the following binary vectors that l would want to create relationship between them : A, B, C and the resulting F
Formulating a relationship between the three:
A ⊕ B = X1
X1 ⊕ C = X2
X2 ⊕ A = F
When first process is done, B and F will be known in the Second process. So we would want to find C and A
F = X2 ⊕ A
= X1 ⊕ C ⊕ A
= A ⊕ B ⊕ C ⊕ A
= B ⊕ C ⊕ 0
Therefore C = F ⊕ B
l cannot find A. Any suggestions on how l can solve this and/or how to create the relationships. where ⊕ is the exclusive or operator
To recap, you have $$\begin{aligned} A \oplus B &= X_1 \\ X_1 \oplus C &= X_2 \\ X_2 \oplus F &= A \end{aligned} \label{NA1}\tag{1}$$ and you have determined that since $$A \oplus B \oplus C \oplus F = A \label{NA2}\tag{2}$$ then $$C = F \oplus B \label{NA3}\tag{3}$$
Thing is, modifying the value of $A$ only affects $X_1$ and $X_2$, which are free variables. For example, consider the case where we simply set $A = 0$, but keep $B$, $C$, and $F$ unchanged: $$\begin{aligned} 0 \oplus B &= X_1^\prime \\ X_1^\prime \oplus C &= X_2^\prime \\ X_2^\prime \oplus F &= 0 \end{aligned}$$ Obviously, $X_1^\prime = B$, and $X_2^\prime = B \oplus C$, which leads to $$B \oplus C \oplus F = 0$$ which is just $\eqref{NA3}$ reordered, $$C = F \oplus B$$ In other words, if $C = F \oplus B$, unless you restrict $X_1$ or $X_2$ somehow, any $A$ including $A = 0$ will fulfill $A \oplus B \oplus C \oplus F = A$.