Tautology/Examples/((not p) implies (q or r)) iff ((not q) implies ((not r) implies p))
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Examples of Tautologies
The WFF of propositional logic:
- $\paren {\paren {\lnot p} \implies \paren {q \lor r} } \iff \paren {\paren {\lnot q} \implies \paren {\paren {\lnot r} \implies p} }$
is a tautology.
Proof
Proof by truth table:
$\begin{array}{cccccc|c|ccccccc} ((\lnot & p) & \implies & (q & \lor & r)) & \iff & ((\lnot & q) & \implies & ((\lnot & r) & \implies & p)) \\ \hline
T & F & F & F & F & F & T & T & F & F & T & F & F & F \\ T & F & T & F & T & T & T & T & F & T & F & T & T & F \\ T & F & T & T & T & F & T & F & T & T & T & F & F & F \\ T & F & T & T & T & T & T & F & T & T & F & T & T & F \\ F & T & T & F & F & F & T & T & F & T & T & F & T & T \\ F & T & T & F & T & T & T & T & F & T & F & T & T & T \\ F & T & T & T & T & F & T & F & T & F & T & F & T & T \\ F & T & T & T & T & T & T & F & T & T & F & T & T & T \\
\end{array}$
As can be seen by inspection, the truth value under the main connective is true for all rows.
$\blacksquare$
Sources
- 1988: Alan G. Hamilton: Logic for Mathematicians (2nd ed.) ... (previous) ... (next): $\S 1$: Informal statement calculus: $\S 1.2$: Truth functions and truth tables: Exercise $4$