I want to find the DNF and CNF of the following expression
$$ x \oplus y \oplus z $$
I tried by using
$$x \oplus y = (\neg x\wedge y) \vee (x\wedge \neg y)$$
but it got all messy.
I also plotted it in Wolfram Alpha, and of course it showed them, but not the steps you need to make to get there.
Any ideas to how this could be done?
Best Answer
Simply write down the truth table, which is quite simple to find, and deduce your CNF and DNF.
\begin{array}{| c | c | c | c |} \hline X & Y & Z & \\ \hline T & T & T & T \\ \hline T & T & F & F \\ \hline T & F & T & F \\ \hline T & F & F & T \\ \hline F & T & T & F \\ \hline F & T & F & T \\ \hline F & F & T & T \\ \hline F & F & F & F \\ \hline \end{array}
If you want to find DNF, you have to look at all rows that ends with $T$. When you find those rows, take the $x, y,$ and $z$ values from each respective column. Thus, you get $$(x \wedge y \wedge z) \vee (x \wedge \neg y \wedge \neg z) \vee (\neg x \wedge y \wedge \neg z) \vee (\neg x \wedge \neg y \wedge z).$$ Similarly, you can find CNF
$$ (\lnot x \lor \lnot y \lor z) \land (\lnot x \lor y \lor \lnot z) \land (x \lor \lnot y \lor \lnot z) \land (x \lor y \lor z) $$