document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1181635>Question 1181635</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Find the truth value of the following: Show the solution.\r<BR>\n" );
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document.write( "4. (q∨r)↔[(¬q→(r∧¬p))]<BR>\n" );
document.write( "5. (¬s↔(r→¬q))↔[(s∨p)∧¬(q∧r)] </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #849985 by </B><A HREF=/tutors/aboutme.mpl?userid=CPhill><B>CPhill(1959)</B></A><img src=\"/images/stars/stars-12.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=CPhill><IMG SRC=http://www.algebra.com/images/aboutme-small.gif BORDER=0 alt=\"About Me\" VALIGN=MIDDLE></A>&nbsp;<IMG SRC=http://www.algebra.com/cgi-bin/show-face.mpl?user=CPhill><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=849985\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Let's analyze the truth values of these compound statements.  We'll use a truth table approach.\r<BR>\n" );
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document.write( "**4. (q∨r)↔[(¬q→(r∧¬p))]**\r<BR>\n" );
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document.write( "First, we need to consider all possible truth values for p, q, and r.  Since there are three variables, there are 2³ = 8 possible combinations.\r<BR>\n" );
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document.write( "| p | q | r | q∨r | ¬q | ¬p | r∧¬p | ¬q→(r∧¬p) | (q∨r)↔[(¬q→(r∧¬p))] |<BR>\n" );
document.write( "|---|---|---|-----|----|----|------|---------|-----------------------|<BR>\n" );
document.write( "| T | T | T |  T  | F  | F  |  F   |    T    |           T           |<BR>\n" );
document.write( "| T | T | F |  T  | F  | F  |  F   |    T    |           T           |<BR>\n" );
document.write( "| T | F | T |  T  | T  | F  |  F   |    F    |           F           |<BR>\n" );
document.write( "| T | F | F |  F  | T  | F  |  F   |    F    |           T           |<BR>\n" );
document.write( "| F | T | T |  T  | F  | T  |  T   |    T    |           T           |<BR>\n" );
document.write( "| F | T | F |  T  | F  | T  |  F   |    T    |           T           |<BR>\n" );
document.write( "| F | F | T |  T  | T  | T  |  T   |    T    |           T           |<BR>\n" );
document.write( "| F | F | F |  F  | T  | T  |  F   |    F    |           T           |\r<BR>\n" );
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document.write( "As you can see from the final column, the statement (q∨r)↔[(¬q→(r∧¬p))] is a **tautology** (always true).\r<BR>\n" );
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document.write( "**5. (¬s↔(r→¬q))↔[(s∨p)∧¬(q∧r)]**\r<BR>\n" );
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document.write( "Now, let's analyze the second statement.  We have four variables (p, q, r, s), so there are 2⁴ = 16 possible truth value combinations.  Constructing the full truth table is a bit lengthy, but we can illustrate the process and the final result.\r<BR>\n" );
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document.write( "| p | q | r | s | ¬s | ¬q | r→¬q | ¬s↔(r→¬q) | s∨p | q∧r | ¬(q∧r) | (s∨p)∧¬(q∧r) | (¬s↔(r→¬q))↔[(s∨p)∧¬(q∧r)] |<BR>\n" );
document.write( "|---|---|---|---|----|----|------|----------|-----|-----|-------|-------------|-----------------------------|<BR>\n" );
document.write( "| T | T | T | T | F  | F  |  F   |     T    |  T  |  T  |   F   |      F      |               F             |<BR>\n" );
document.write( "| ... (14 more rows) ... | ... | ... | ... | ... | ... | ... | ... | ... | ... | ... | ... | ... |<BR>\n" );
document.write( "| F | F | F | F | T  | T  |  T   |     T    |  F  |  F  |   T   |      F      |               F             |\r<BR>\n" );
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document.write( "(The table is truncated for brevity.  You would fill in all 16 rows.)\r<BR>\n" );
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document.write( "After completing the full truth table (which I recommend you do to verify), you will find that the final column has both T and F values. This means the statement is a **contingency**—its truth value depends on the truth values of its variables. It is *not* a tautology or a contradiction.\r<BR>\n" );
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document.write( "**In summary:**\r<BR>\n" );
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document.write( "4.  (q∨r)↔[(¬q→(r∧¬p))] is a **tautology** (always true).<BR>\n" );
document.write( "5.  (¬s↔(r→¬q))↔[(s∨p)∧¬(q∧r)] is a **contingency** (sometimes true, sometimes false).<BR>\n" );
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