Logical Equivalence Help

**triathlete** · February 2nd 2010, 12:30 PM

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

I can get a few steps into it, then get stuck. Please help!

**Plato** · February 2nd 2010, 12:46 PM

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

I can get a few steps into it, then get stuck. Please help!

$\begin{gathered} \left( {p \to q} \right) \vee \left( {p \to r} \right) \vee p \hfill \\ \left( {\neg p \vee q} \right) \vee \left( {\neg p \vee r} \right) \vee p \hfill \\ \neg p \vee \left( {q \vee r} \right) \vee p \hfill \\ \left( {\neg p \vee p} \right) \vee \left( {q \vee r} \right) \hfill \\ \end{gathered}$

**novice** · February 2nd 2010, 01:06 PM

By predicate calculus:

$P$ (hypothesis)

$P\vee (P\rightarrow R)$ (Disjunction Introduction)

$(P\vee (P\rightarrow R))\vee (P\rightarrow Q)$ (Disjunction Introduction and end of proof)

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