me gustaria que me corrijjjgan con respecto a este ejercicio sobre todo en las propiedades
\( \[sean:p,q,r\,proposiciones\,\logicas\,y\,considere\,la\,nueva\,proposicion:\] \)
\( \[\left[ {\left( {p \vee q} \right) \Leftrightarrow \left( {p \wedge r} \right)} \right] \Rightarrow \left[ {\left( {q \Rightarrow p} \right) \wedge \left( {p \Rightarrow r} \right)} \right]\,\left( * \right)\]
\)
\( \[demostrar\,que\,\left( * \right)\,es\,tautologia\] \)
mi desarrollo:
\(
\[\left[ {\left[ {\left( {p \vee q} \right) \Rightarrow \left( {p \wedge r} \right)} \right] \wedge \left[ {\left( {p \wedge r} \right) \Rightarrow \left( {p \vee q} \right)} \right]} \right] \Rightarrow \left[ {\left( {q \Rightarrow r} \right)} \right]/\left( {transitividad\,de\, \Rightarrow } \right)\] \)
\( \[\left[ {\left[ {\left( {p \vee q} \right) \Rightarrow \left( {p \wedge r} \right)} \right] \wedge \left[ {\left( {p \wedge r} \right) \Rightarrow \left( {p \vee q} \right)} \right]} \right]\] \) (definicion de <==>)
\( \[\left[ {\left[ {\overline {\left( {p \vee q} \right)} \vee \left( {p \wedge r} \right)} \right] \wedge \left[ {\overline {\left( {p \wedge r} \right)} \vee \left( {p \vee q} \right)} \right]} \right]/\left( {\left( {\overline a \vee b} \right) \equiv a \Rightarrow b} \right)\] \)
\( \[\left[ {\left[ {\left( {\overline p \wedge \overline q } \right) \vee \left( {p \wedge r} \right)} \right] \wedge \left[ {\left( {\overline p \vee \overline r } \right) \vee \left( {p \vee q} \right)} \right]} \right]/\left( {ley\,de\,morgan} \right)\]
\)
\(
\[\left[ {\left[ {\left( {\overline q } \right) \vee \left( r \right)} \right] \wedge \left[ {\left( {\overline p \vee \overline r } \right) \vee \left( {p \vee q} \right)} \right]} \right]/\left( {Tauto\log ia\,de\,a \wedge b \Rightarrow a} \right)\] \)
\( \[\left[ {\left[ {\left( {\overline q } \right) \vee \left( r \right)} \right] \wedge \left[ {\left( {\overline p \vee p} \right) \vee \left( {\overline r \vee q} \right)} \right]} \right]/\left( {distributividad\,del\, \vee } \right)\] \)
\( \[\left[ {\left[ {\left( {\overline q } \right) \vee \left( r \right)} \right] \wedge \left[ {T \vee \left( {\overline r \vee q} \right)} \right]} \right]/\left( {\overline a \vee a \Rightarrow T} \right)\] \)
\( \[\left[ {\left[ {\left( {\overline q } \right) \vee \left( r \right)} \right] \wedge \left[ T \right]} \right]/\left( {T \wedge a \Rightarrow a} \right)\] \)
\( \[\left[ {\left( {\overline q } \right) \vee \left( r \right)} \right]/\left( {\left( {\overline a \vee b} \right) \equiv a \Rightarrow b} \right)\] \)
\( \[q \Rightarrow r\] \)
muchas gracias de antemano
saludos!