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\n" ); document.write( "T = true
\n" ); document.write( "F = false\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "We have three variables p, q and r
\n" ); document.write( "We need exactly
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "The first column for p will have four T's in a row followed by four F's in a row.
\n" ); document.write( "The second column for q will have two copies of TTFF
\n" ); document.write( "The third column for r will have four copies of TF
\n" ); document.write( "This will guarantee to capture every possible outcome\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "We have this table so far
| p | q | r |
| T | T | T |
| T | T | F |
| T | F | T |
| T | F | F |
| F | T | T |
| F | T | F |
| F | F | T |
| F | F | F |
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "Next is to add the column titled ~q which will just flip whatever is in column q
| p | q | r | ~q |
| T | T | T | F |
| T | T | F | F |
| T | F | T | T |
| T | F | F | T |
| F | T | T | F |
| F | T | F | F |
| F | F | T | T |
| F | F | F | T |
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "Do a similar operation for column ~r
| p | q | r | ~q | ~r |
| T | T | T | F | F |
| T | T | F | F | T |
| T | F | T | T | F |
| T | F | F | T | T |
| F | T | T | F | F |
| F | T | F | F | T |
| F | F | T | T | F |
| F | F | F | T | T |
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "Let's add the p ^ ~q column to the table we're building up
| p | q | r | ~q | ~r | p ^ ~q |
| T | T | T | F | F | F |
| T | T | F | F | T | F |
| T | F | T | T | F | T |
| T | F | F | T | T | T |
| F | T | T | F | F | F |
| F | T | F | F | T | F |
| F | F | T | T | F | F |
| F | F | F | T | T | F |
\n" ); document.write( "Recall that p ^ q is only true when p,q are true at the same time; otherwise, it's false.\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "Next, we can form the q v ~r column
| p | q | r | ~q | ~r | p ^ ~q | q v ~r |
| T | T | T | F | F | F | T |
| T | T | F | F | T | F | T |
| T | F | T | T | F | T | T |
| T | F | F | T | T | T | T |
| F | T | T | F | F | F | T |
| F | T | F | F | T | F | T |
| F | F | T | T | F | F | F |
| F | F | F | T | T | F | T |
\n" ); document.write( "The disjunction p v q is only false when p,q are both false; otherwise, it's true.\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "And finally, here is the table that you'll have as the final answer.
| p | q | r | ~q | ~r | p ^ ~q | q v ~r | (q v ~r) -> (p ^ ~q) |
| T | T | T | F | F | F | T | F |
| T | T | F | F | T | F | T | F |
| T | F | T | T | F | T | T | T |
| T | F | F | T | T | T | T | T |
| F | T | T | F | F | F | T | F |
| F | T | F | F | T | F | T | F |
| F | F | T | T | F | F | F | T |
| F | F | F | T | T | F | T | F |
\n" ); document.write( "It's definitely a lot to take in, especially because of the 8 rows to keep track of everything.
\n" ); document.write( "For the last column, the statement p -> q is only false when p = T and q = F; otherwise, the conditional statement is true.\r
\n" ); document.write( "\n" ); document.write( "
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