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If $(\cdot)$ is the bitwise XOR operation and $X$ and $Y$ are any two binary numbers, then $Z$ is the bitmask such that:

\begin{align}X(\cdot)Y&=Z \\ X(\cdot)Z&=Y\\ Y(\cdot)Z&=X \end{align}

Now how do I prove that:

For some $W$ (a binary number) such that $W\neq Z$ , $$X(\cdot)W=Y$$ does not exist?

lioness99a
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juztcode
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  • Because "$=$" is an equivalence relation... so that $Z=X(.)Y=W$. – WhatsUp Mar 03 '20 at 11:43
  • @WhatsUp , sorry the question earlier was a mistake, please take a look I edited it – juztcode Mar 03 '20 at 11:44
  • It's not much different. All you need to realize is what you've written: if $X(.)Y=Z$ then $X(.)Z=Y$. Hence if $X(.)W=Y$ then $X(.)Y=W$ and therefore $Z=X(.)Y=W$. – WhatsUp Mar 03 '20 at 11:48

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