- #1
evinda
Gold Member
MHB
- 3,741
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Hello! (Wave)
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? :unsure:
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? :unsure:
