# Truth table

• October 17th 2009, 01:46 PM
zap1231
Truth table
Use a truth table to show that (p->q)-> r and p->(q->r) are or are not logically equpvalent
• October 17th 2009, 01:53 PM
tonio
Quote:

Originally Posted by zap1231
Use a truth table to show that (p->q)-> r and p->(q->r) are or are not logically equpvalent

No need for a whole talbe: when you give the value F to all p,q,r, you get in (p --> q) --> r the value F, whereas in p --> (q --> r) you get T.

Tonio
• October 17th 2009, 05:27 PM
Soroban
Hello, zap1231!

Quote:

Use a truth table to show that $(p \to q) \to r$ and $p \to (q\to r)$
are or are not logically equivalent

. . $\begin{array}{|c|c|c||}
p & q & r \\ \hline
T & T & T \\ T & T & F \\ T & F & T \\ T & F & F \\
F & T & T \\ F & T & F \\ F & F & T \\ F & F & F\\ \hline & & \\ \end{array}$
$
\begin{array}{cccccc} [(p & \to & q) & \to & r] & \Longleftrightarrow \\ \hline
T & T & T & T & T & T \\ T & T & T & F & F & T\\ T & F & F & T & T & T \\
T & F & F & T & F & T \\ F & T & T & T & T & T \\ F & T & T & F & F & {\color{red}F} \\
F & T & F & T & T & T \\ F & T & F & F & F & {\color{red}F} \\\hline 1 & 2 & 1 & 3 & 1 & 4
\end{array}$
$\begin{array}{ccccc} [p & \to & (q & \to & r)] \\ \hline
T & T & T & T & T \\ T & F & T & F & F \\ T & T & F & T & T \\
T & T & F & T & F \\ F & T & T & T & T \\ F & T & T & F & F \\
F & T & F & T & T \\ F & T & F & T & F \\ \hline 1 & 3 & 1 & 2 & 1 \end{array}
$

. . . . . . . . . . . . . . . . . . . . . . . ${\color{blue}\uparrow}$
. . . . . . . . . . . . . . . . . . .not equivalent