document.write( "Question 162436This question is from textbook
\n" ); document.write( ": P v (Q & R) <=> (P v Q) & (P v R)
\n" ); document.write( " \n" ); document.write( "
Algebra.Com's Answer #119710 by Edwin McCravy(20060)\"\" \"About 
You can put this solution on YOUR website!
\r\n" );
document.write( "P v (Q & R) <=> (P v Q) & (P v R)\r\n" );
document.write( "\r\n" );
document.write( "This is the distributive law of v over &.\r\n" );
document.write( "\r\n" );
document.write( "Using the rules:\r\n" );
document.write( "\r\n" );
document.write( "Under P put TTTTFFFF,\r\n" );
document.write( "\r\n" );
document.write( "Under Q put TTFFTTFF,\r\n" );
document.write( "\r\n" );
document.write( "Under R put TFTFTFTF,\r\n" );
document.write( "\r\n" );
document.write( "The rule for \"~\" (not) is \"~T is F and ~F is T\",\r\n" );
document.write( "\r\n" );
document.write( "The rule for \"&\" (and) is \"only T&T is T, all others F\",\r\n" );
document.write( "\r\n" );
document.write( "The rule for \"v\" (or) is \"only FVF is F, all others T\",\r\n" );
document.write( "\r\n" );
document.write( "The rule for \"->\" (if..then...) is \"only T->F is F, all other T\",\r\n" );
document.write( "\r\n" );
document.write( "The rule for \"<->\" (biconditional \"the same\") is \r\n" );
document.write( "\"only T<->T and F<->F are T, all others F,\r\n" );
document.write( "\r\n" );
document.write( "make this truth table for P v (Q & R) <-> (P v Q) & (P v R)\r\n" );
document.write( "\r\n" );
document.write( "|P|Q|R|~Q|Q&R|Pv(Q&R)|PvQ|PvR|(PvQ)&(PvR)|Pv(Q&R)<->(PvQ)&(PvR)|\r\n" );
document.write( "|T|T|T| F| T |  T    | T | T |     T     |        T            |\r\n" );
document.write( "|T|T|F| F| F |  T    | T | T |     T     |        T            |\r\n" );
document.write( "|T|F|T| T| F |  T    | T | T |     T     |        T            |\r\n" );
document.write( "|T|F|F| T| F |  T    | T | T |     T     |        T            |\r\n" );
document.write( "|F|T|T| F| T |  T    | T | T |     T     |        T            |\r\n" );
document.write( "|F|T|F| F| F |  F    | T | F |     F     |        T            |\r\n" );
document.write( "|F|F|T| T| F |  F    | F | T |     F     |        T            |\r\n" );
document.write( "|F|F|F| T| F |  F    | F | F |     F     |        T            |\r\n" );
document.write( "\r\n" );
document.write( "The proposition is proved because there are only T's in the last \r\n" );
document.write( "column.\r\n" );
document.write( "\r\n" );
document.write( "Therefore we can replace the biconditional symbol <->, by the\r\n" );
document.write( "stronger equivalence symbol <=> and write\r\n" );
document.write( "\r\n" );
document.write( "P v (Q & R) <=> (P v Q) & (P v R)\r\n" );
document.write( "\r\n" );
document.write( "Edwin
\n" ); document.write( "
\n" );