Logical Equivalence Help

• Feb 2nd 2010, 12:30 PM
triathlete
Logical Equivalence Help
Hello, I have a problem in my computer science class that goes as follows:

" prove the following using logical proofs (not truth tables)"

(p --> q) V (p --> r) V p is a tautology

• Feb 2nd 2010, 12:46 PM
Plato
Quote:

Originally Posted by triathlete
Hello, I have a problem in my computer science class that goes as follows:

" prove the following using logical proofs (not truth tables)"

(p --> q) V (p --> r) V p is a tautology

$\displaystyle \begin{gathered} \left( {p \to q} \right) \vee \left( {p \to r} \right) \vee p \hfill \\ \left( {\neg p \vee q} \right) \vee \left( {\neg p \vee r} \right) \vee p \hfill \\ \neg p \vee \left( {q \vee r} \right) \vee p \hfill \\ \left( {\neg p \vee p} \right) \vee \left( {q \vee r} \right) \hfill \\ \end{gathered}$
$\displaystyle P$ (hypothesis)
$\displaystyle P\vee (P\rightarrow R)$ (Disjunction Introduction)
$\displaystyle (P\vee (P\rightarrow R))\vee (P\rightarrow Q)$ (Disjunction Introduction and end of proof)