0
$\begingroup$

How can:

$\lnot(x \lor y \lor z)=\lnot x \land \lnot y \land \lnot z$

be obtained from:

$\lnot (x \land y \land z) = \lnot x \lor \lnot y \lor \lnot z$   

such that $x, y, z $ are logical variables.

I have tried some manipulation of the equations such as:

\begin{align*} \lnot (x \land y \land z) &= \lnot x \lor \lnot z \lor \lnot y \\ \lnot x \lor \lnot (y \land z) &= \lnot(x \land y \land z) \\ \lnot x \lor \lnot y \lor \lnot z &= \lnot(x \land y \land z) \\ \end{align*}

However this doesn't seem to be the right direction to go.

$\endgroup$
1
$\begingroup$

You can prove the result by taking advantage of the fact that double negation is the identity function. \begin{align*} \lnot x \land \lnot y \land \lnot z &= \lnot\lnot(\lnot x \land \lnot y \land \lnot z) && \text{(by law of double negation)}\\ &= \lnot[\lnot(\lnot x \land \lnot y \land \lnot z)] \\ &= \lnot[\lnot(\lnot x) \lor \lnot(\lnot y) \lor \lnot(\lnot z)] && \text{(by the other of DeMorgan's Laws)} \\ &= \lnot[x \lor y \lor z] && \text{(by law of double negation)}\\ \end{align*}

Thus, $\lnot[x \lor y \lor z] = \lnot x \land \lnot y \land \lnot z$

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.