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\n" ); document.write( "Answer: true\r
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\n" ); document.write( "\n" ); document.write( "Explanation\r
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\n" ); document.write( "\n" ); document.write( "Here is a review of various truth table rules
\n" ); document.write( "https://www.algebra.com/algebra/homework/Conjunction/truth-table1.lesson\r
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\n" ); document.write( "\n" ); document.write( "Since p = false, this makes ~p = true
\n" ); document.write( "Apply the conjunction ~p ^ q to get true ^ true = true
\n" ); document.write( "The ~p ^ q portion is true.
\n" ); document.write( "So is ~r because r = false and ~r = true\r
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\n" ); document.write( "\n" ); document.write( "Both (~p ^ q) and ~r produce the same truth values of \"true\"
\n" ); document.write( "Overall, the statement (~p ^ q) <--> ~r is true when p = false, q = true, r = false\r
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\n" ); document.write( "\n" ); document.write( "If you wanted, you could make a full truth table like so.
\n" ); document.write( "The row to highlight or focus on would be row 6 marked in red.
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| p | q | r | ~p | ~r | ~p ^ q | (~p ^ q) <--> ~r |
| T | T | T | F | F | F | T |
| T | T | F | F | T | F | F |
| T | F | T | F | F | F | T |
| T | F | F | F | T | F | F |
| F | T | T | T | F | T | F |
| F | T | F | T | T | T | T |
| F | F | T | T | F | F | T |
| F | F | F | T | T | F | F |
\n" ); document.write( "and this is what the condensed table looks like when focusing on row 6 only
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| p | q | r | ~p | ~r | ~p ^ q | (~p ^ q) <--> ~r |
| F | T | F | T | T | T | T |
\n" ); document.write( "I used spreadsheet software to generate the truth tables quickly.
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