Truth Table - implies false I'm work with a task where I am not exactly sure if I proceed right. The task is saying: "We define the operation $\oplus$  by $a \oplus b = (a \wedge \neg b) \vee (\neg a \wedge b)$. Give the truth table of:" 

$$(a \oplus b) \to \text{False}$$

So my question is, do I answer this with a truth table like this:



Or do I answer like this: 



Or is it something I am not aware of? Thanks for any answer! :D
"False" is not a variable, so you should not add it into the truth table. It makes no sense to say "what if false is true", which is exactly what the second row in your first table asks.

Your second table is correct.
$G\equiv[(A\land\lnot B)\lor(\lnot A\land B)]$
$====================$
$C\equiv[\lnot B]$
$D\equiv[A\land C]$
$E\equiv[\lnot A]$
$F\equiv[E\land B]$
$G\equiv[D\lor F]$    

$\begin{array}{cc|cccc|c}
A&amp;B&amp;C&amp;D&amp;E&amp;F&amp;G\\
\hline
0&amp;0&amp;1&amp;0&amp;1&amp;0&amp;0\\
0&amp;1&amp;0&amp;0&amp;1&amp;1&amp;1\\
1&amp;0&amp;1&amp;1&amp;0&amp;0&amp;1\\
1&amp;1&amp;0&amp;0&amp;0&amp;0&amp;0
\end{array}$