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- #1

- Apr 13, 2013

- 3,718

I want to calculate the truth tables of the following propositions:

$$(p \land q) \lor (\lnot p \land q) \to q \\ p \land \lnot q \to r$$

I have done the following:

\begin{equation*}

\begin{array}{c|c|c|c|c}

p & q & p \land q & \lnot p \land q & (p \land q) \lor (\lnot p \land q) \to q \\

\hline

1 & 1 & 1 & 0 & 1 \\

1 & 0 & 0 & 0 & 1 \\

0 & 0 & 0 & 0 & 1 \\

0 & 1 & 0 & 1 & 1

\end{array}

\end{equation*}

and

\begin{equation*}

\begin{array}{c|c|c|c}

p & q & \lnot q & p \land \lnot q \\

\hline

1 & 1 & 0 & 0 \\

1 & 0 & 1 & 1 \\

0 & 0 & 1 & 0 \\

0 & 1 & 0 & 0

\end{array}

\end{equation*}

If $p=q=1$ and $r=0$, then $p \land \lnot q \to r$ is true, and the same holds if $r=1$. The same holds when $p=q=0$ and $p=0$, $q=1$.

If $p=1$ and $q=0$, then if $r=0$ then $p \land \lnot q \to r$ is false, and if $r=1$ then it is true.

Is everything right? Or have I done something wrong?