document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1174481>Question 1174481</A>: <I><SPAN STYLE=\"word-break: break-all;\"> TRUTH TABLES: Determine the validity of the argument using truth tables. \r<BR>\n" );
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document.write( "A polygon is regular or a polygon has a side which is longer than another side of the polygon.\r<BR>\n" );
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document.write( "If a polygon is regular then all the sides of the polygon are congruent.\r<BR>\n" );
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document.write( "If a polygon has one side which is longer than another side of the polygon, then an interior angle of the polygon has measure greater than one of the interior angles of the polygon.\r<BR>\n" );
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document.write( "Therefore, either the polygon have congruent sides or an interior angle of the polygon has a measure greater than one of the interior angles of the polygon.\r<BR>\n" );
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document.write( "SOLUTIONS AND ANSWERS please.\r<BR>\n" );
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document.write( "THANK YOU </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #850642 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=850642\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Let's break down this argument into propositional logic and use a truth table to determine its validity.\r<BR>\n" );
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document.write( "**1. Define Propositions:**\r<BR>\n" );
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document.write( "* **P:** A polygon is regular.<BR>\n" );
document.write( "* **Q:** A polygon has a side which is longer than another side of the polygon.<BR>\n" );
document.write( "* **R:** All the sides of the polygon are congruent.<BR>\n" );
document.write( "* **S:** An interior angle of the polygon has a measure greater than one of the interior angles of the polygon.\r<BR>\n" );
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document.write( "**2. Express the Argument in Symbolic Form:**\r<BR>\n" );
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document.write( "* Premise 1: P ∨ Q<BR>\n" );
document.write( "* Premise 2: P → R<BR>\n" );
document.write( "* Premise 3: Q → S<BR>\n" );
document.write( "* Conclusion: R ∨ S\r<BR>\n" );
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document.write( "**3. Construct the Truth Table:**\r<BR>\n" );
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document.write( "We need to consider all possible combinations of truth values for P, Q, R, and S.\r<BR>\n" );
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document.write( "| P | Q | R | S | P ∨ Q | P → R | Q → S | R ∨ S |<BR>\n" );
document.write( "|---|---|---|---|-------|-------|-------|-------|<BR>\n" );
document.write( "| T | T | T | T | T     | T     | T     | T     |<BR>\n" );
document.write( "| T | T | T | F | T     | T     | F     | T     |<BR>\n" );
document.write( "| T | F | T | T | T     | T     | T     | T     |<BR>\n" );
document.write( "| T | F | T | F | T     | T     | T     | T     |<BR>\n" );
document.write( "| F | T | F | T | T     | T     | T     | T     |<BR>\n" );
document.write( "| F | T | F | F | T     | T     | F     | F     |<BR>\n" );
document.write( "| F | F | F | T | F     | T     | T     | T     |<BR>\n" );
document.write( "| F | F | F | F | F     | T     | T     | F     |\r<BR>\n" );
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document.write( "**4. Evaluate the Argument:**\r<BR>\n" );
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document.write( "To determine validity, we need to check if the conclusion (R ∨ S) is true whenever all the premises (P ∨ Q, P → R, Q → S) are true.\r<BR>\n" );
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document.write( "Let's examine the rows where all premises are true:\r<BR>\n" );
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document.write( "* **Row 1:** P, Q, R, S are all true. All premises and the conclusion are true.<BR>\n" );
document.write( "* **Row 3:** P, R, S are true; Q is false. All premises and the conclusion are true.<BR>\n" );
document.write( "* **Row 4:** P, R are true; Q, S are false. All premises and the conclusion are true.<BR>\n" );
document.write( "* **Row 5:** Q, S are true; P, R are false. All premises and the conclusion are true.<BR>\n" );
document.write( "* **Row 7:** S is true; P, Q, R are false. Premise 2 and 3 are true because the antecedent is false. Premise 1 is false. The conclusion is true.<BR>\n" );
document.write( "* **Row 8:** All are false. Premise 2, and 3 are true because the antecedent is false. Premise 1 is false. The conclusion is false.\r<BR>\n" );
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document.write( "Let's adjust the table to show the combined premises. We only care about rows where all premises are true.\r<BR>\n" );
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document.write( "| P | Q | R | S | P ∨ Q | P → R | Q → S | (P ∨ Q) ∧ (P → R) ∧ (Q → S) | R ∨ S |<BR>\n" );
document.write( "|---|---|---|---|-------|-------|-------|-------------------------------|-------|<BR>\n" );
document.write( "| T | T | T | T | T     | T     | T     | T                             | T     |<BR>\n" );
document.write( "| T | F | T | T | T     | T     | T     | T                             | T     |<BR>\n" );
document.write( "| T | F | T | F | T     | T     | T     | T                             | T     |<BR>\n" );
document.write( "| F | T | F | T | T     | T     | T     | T                             | T     |<BR>\n" );
document.write( "| F | F | F | T | F     | T     | T     | F                             | T     |<BR>\n" );
document.write( "| F | F | F | F | F     | T     | T     | F                             | F     |<BR>\n" );
document.write( "| F | T | F | F | T     | T     | F     | F                             | F     |<BR>\n" );
document.write( "| T | T | T | F | T     | T     | F     | F                             | T     |\r<BR>\n" );
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document.write( "In every row where all premises are true, the conclusion is also true. Therefore, the argument is valid.<BR>\n" );
document.write( " </SPAN>\n" );
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