Checkpoint 3.4.1.
Use truth tables to prove that the statements below are tautologies.
- \(\left [ (p \implies q) \land p \right ] \implies q\)
- \(\left [ (p \land (\lnot p) \right ] \implies q\)
Section 3.4 Exercises
Checkpoint 3.4.2.
Let \(T\) be a tautology (a statement that always has truth value \(T\)). Show that for any proposition \(p\text{,}\) \(T \lor p \equiv T\) and \(T \land p \equiv p\text{.}\)
Checkpoint 3.4.3.
If \(F\) represents a contradiction (a statement that always has truth value \(F\)) and \(p\) is any proposition, what can be said about \(F \land p\) and \(F\lor p\text{?}\)
Checkpoint 3.4.4.
Prove that for all statements \(p\text{,}\) \(q\) and \(r\text{,}\)
\begin{equation*}
\left[p \implies (q \lor r) \right ] \equiv \left [ (p \land (\lnot q) ) \implies r \right ]
\end{equation*}
by (a) using a truth table and (b) stringing together previously established logical equivalences (be sure to cite each equivalence law you use).
Checkpoint 3.4.5.
The previous problem shows also that \([\left [ (p \land (\lnot q) ) \implies r \right ] \equiv \left [ (p \land (\lnot r) ) \implies q \right ]\text{.}\) Explain why this means you only need to prove one of those statements (not both!) to prove the original implication.
