document.write( "Question 478096: thank you.\r
\n" ); document.write( "\n" ); document.write( "6. Determine which, if any, of the three statements are equivalent.
\n" ); document.write( "I) If the pipe is leaking, then I will not call the roofer.
\n" ); document.write( "II) Either the pipe is leaking or I will call the roofer.
\n" ); document.write( "III) If the pipe is not leaking, then I will call the roofer.
\n" ); document.write( "
\n" ); document.write( " I and II are equivalent
\n" ); document.write( " II and III are equivalent
\n" ); document.write( " I and III are equivalent
\n" ); document.write( " I, II, and III are equivalent
\n" ); document.write( " None are equivalent \n" ); document.write( "

Algebra.Com's Answer #327621 by Theo(13342) $\"\"$ $\"About$

You can put this solution on YOUR website!
this is a scary one but i think i have the answer.
\n" ); document.write( "statements are equivalent if they have the same truth value for all possible conditions.
\n" ); document.write( "you need to create a truth table for each statement.
\n" ); document.write( "the statements are:
\n" ); document.write( "I) If the pipe is leaking, then I will not call the roofer.
\n" ); document.write( "II) Either the pipe is leaking or I will call the roofer.
\n" ); document.write( "III) If the pipe is not leaking, then I will call the roofer.
\n" ); document.write( "the symbols we will use are:
\n" ); document.write( "p = the pipe is leaking
\n" ); document.write( "q = i will call the roofer.
\n" ); document.write( "converting the statements into symbols gets:
\n" ); document.write( "statement I becomes:
\n" ); document.write( "p->~q
\n" ); document.write( "statement II becomes:
\n" ); document.write( "(p)v(q)
\n" ); document.write( "statement III becomes:
\n" ); document.write( "~p->q
\n" ); document.write( "to see if the statements are equivalent, we use truth tables.
\n" ); document.write( "the truth tables for these statements are:
\n" ); document.write( "

\r\n" );
document.write( "       p   q   ~p   ~q   p->~q   (p)v(q)   ~p->q\r\n" );
document.write( "       T   T    F    F       F        T        T\r\n" );
document.write( "       T   F    F    T       T        T        T\r\n" );
document.write( "       F   T    T    F       T        T        T\r\n" );
document.write( "       F   F    T    T       T        F        F\r\n" );
document.write( "

\n" ); document.write( "contrary to what you might believe otherwise, the truth table points to the fact that (p)v(q) is equivalent to ~p->q because their truth tables are the same under all conditions.
\n" ); document.write( "this makes the statements equivalent.
\n" ); document.write( "The answer to the question is that statements II and III are logically equivalent.
\n" ); document.write( "a reference on equivalent statements is shown below:
\n" ); document.write( "http://it.edgecombe.edu/homepage/killorant/MAT140/Module2/EquivalentStatements.pdf
\n" ); document.write( "-> means \"if then\"
\n" ); document.write( "v means \"or\"
\n" ); document.write( " \n" ); document.write( "

\n" );