Proving Other Kinds of Claims

$\newcommand{\N}{\mathbb N} \newcommand{\Z}{\mathbb Z} \newcommand{\Q}{\mathbb Q} \newcommand{\R}{\mathbb R} \newcommand{\lt}{<} \newcommand{\gt}{>} \newcommand{\amp}{&} \definecolor{fillinmathshade}{gray}{0.9} \newcommand{\fillinmath}[1]{\mathchoice{\colorbox{fillinmathshade}{$\displaystyle \phantom{\,#1\,}$}}{\colorbox{fillinmathshade}{$\textstyle \phantom{\,#1\,}$}}{\colorbox{fillinmathshade}{$\scriptstyle \phantom{\,#1\,}$}}{\colorbox{fillinmathshade}{$\scriptscriptstyle\phantom{\,#1\,}$}}} $

Subsection 3.2.5 Proving Other Kinds of Claims

So far, we’ve used truth tables to prove claims of the form:

( Premise 1 ∧ Premise 2 ∧ … Premise n ) → Conclusion

We do that by showing that such a claim is a tautology.

But we can also use truth tables to prove the correctness of other kinds of claims (again by showing that they are tautologies). For example, we might want to prove that two logical expressions are equivalent.

Prove that these two logical expressions are equivalent (i.e., for any assignment of truth values to the propositions, either both expressions are true or both are false):

$P \wedge \neg Q \neg (P \rightarrow Q)$

To do this, we will prove that the following claim is a tautology:

$(P \wedge Q)  ((P  Q)) $

Checkpoint 3.2.1.

1. Prove that (( p ∧ q ) → r ) ≡ ( p → ( q → r )). Use the truth table app.