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<!DOCTYPE HTML PUBLIC "-//W3C//DTD HTML 4.01//EN" "http://www.w3.org/TR/html4/strict.dtd"> <html> <head> <title>page_194</title> <link rel="stylesheet" href="reset.css" type="text/css" media="all"> <meta http-equiv="Content-Type" content="text/html; charset=UTF-8" /> </head> <body> <table summary="top nav" border="0" width="100%"> <tr> <td align="left" width="30%" style="background: #EEF3E2"><a style="color: blue; font-size: 120%; font-weight: bold; text-decoration: none; font-family: verdana;" href="page_193.html">< previous page</a></td> <td align="center" width="40%" style="background: #EEF3E2"><strong style="color: #2F4F4F; font-size: 120%;">page_194</strong></td> <td align="right" width="30%" style="background: #EEF3E2"><a style="color: blue; font-size: 120%; font-weight: bold; text-decoration: none; font-family: verdana;" href="page_195.html">next page ></a></td> </tr> <tr> <td align="left" colspan="3" style="background: #ffffff; padding: 20px;"> <table border="0" width="100%" cellpadding="0"><tr><td align="center"> <table border="0" cellpadding="2" cellspacing="0" width="100%"><tr><td align="left"></td> <td align="right"></td> </tr></table></td></tr><tr><td align="left"><p></p><table border="0" cellspacing="0" cellpadding="0" width="100%"><tr><td align="right"><font face="Times New Roman, Times, Serif" size="2" color="#FF0000">Page 194</font></td></tr></table><table border="0" cellspacing="0" cellpadding="0"><tr><td rowspan="5"></td> <td colspan="3" height="12"></td> <td rowspan="5"></td></tr><tr><td colspan="3"></td></tr><tr><td></td> <td><font face="Times New Roman, Times, Serif" size="3">The following pairs of expressions are equivalent:</font></td><td></td></tr><tr><td colspan="3"></td></tr><tr><td colspan="3" height="1"></td></tr></table><table cellpadding="0" cellspacing="0" border="0" width="100%"><tr><td height="12"></td></tr><tr><td><table cellspacing="0" width="326" cellpadding="7"><tr><td valign="top"><font face="Times New Roman, Times, Serif" size="2"><i>Expression</i></font></td><td valign="top"><font face="Times New Roman, Times, Serif" size="2"><i>Equivalent Expression</i></font></td></tr><tr><td valign="top"><font face="Courier New, Courier, Mono New, Courier, Mono" size="2">!(a聽==聽b)</font></td><td valign="top"><font face="Courier New, Courier, Mono New, Courier, Mono" size="2">a聽!=聽b</font></td></tr><tr><td valign="top"><font face="Courier New, Courier, Mono New, Courier, Mono" size="2">!(a聽==聽b聽||聽a聽==聽c)</font></td><td valign="top"><font face="Courier New, Courier, Mono New, Courier, Mono" size="2">a聽!==聽b聽&&聽a聽!=聽c</font></td></tr><tr><td valign="top"><font face="Courier New, Courier, Mono New, Courier, Mono" size="2">!(a聽==聽b聽&&聽c聽>聽d)</font></td><td valign="top"><font face="Courier New, Courier, Mono New, Courier, Mono" size="2">a聽!=b聽||聽c聽<=聽d</font></td></tr></table></td></tr></table><br /><table border="0" cellspacing="0" cellpadding="0"><tr><td rowspan="5"></td> <td colspan="3" height="12"></td> <td rowspan="5"></td></tr><tr><td colspan="3"></td></tr><tr><td></td> <td><font face="Times New Roman, Times, Serif" size="3">Take a close look at these expressions to be sure you understand why they are equivalent. It may help to try evaluating them with some values for a, b, c, and d. Notice the pattern here: The expression on the left is just the one to its right with ! added and the relational and logical operators reversed (for example, == instead of ! = and 娄娄 instead of &&). Remember this pattern. It allows you to rewrite expressions in the simplest form.*</font></td><td></td></tr><tr><td colspan="3"></td></tr><tr><td colspan="3" height="1"></td></tr></table><table border="0" cellspacing="0" cellpadding="0"><tr><td rowspan="5"></td> <td colspan="3" height="12"></td> <td rowspan="5"></td></tr><tr><td colspan="3"></td></tr><tr><td></td> <td><font face="Times New Roman, Times, Serif" size="3">Logical operators can be applied to the results of comparisons. They also can be applied directly to variables of type Boolean. For example, instead of writing</font></td><td></td></tr><tr><td colspan="3"></td></tr><tr><td colspan="3" height="1"></td></tr></table><table border="0" cellspacing="0" cellpadding="0"><tr><td rowspan="5"></td> <td colspan="3" height="12"></td> <td rowspan="5"></td></tr><tr><td colspan="3"></td></tr><tr><td></td> <td><font face="Courier New, Courier, Mono New, Courier, Mono" size="2">isElector聽=聽(age聽>>=聽18聽&&聽district聽==聽23);</font></td><td></td></tr><tr><td colspan="3"></td></tr><tr><td colspan="3" height="1"></td></tr></table><table border="0" cellspacing="0" cellpadding="0"><tr><td rowspan="5"></td> <td colspan="3" height="12"></td> <td rowspan="5"></td></tr><tr><td colspan="3"></td></tr><tr><td></td> <td><font face="Times New Roman, Times, Serif" size="3">to assign a value to Boolean variable </font><font face="Courier New, Courier, Mono New, Courier, Mono" size="3">isElector</font><font face="Times New Roman, Times, Serif" size="3">, we could use two intermediate Boolean variables, </font><font face="Courier New, Courier, Mono New, Courier, Mono" size="3">isVoter</font><font face="Times New Roman, Times, Serif" size="3"> and </font><font face="Courier New, Courier, Mono New, Courier, Mono" size="3">isConstituent</font><font face="Times New Roman, Times, Serif" size="3">:</font></td><td></td></tr><tr><td colspan="3"></td></tr><tr><td colspan="3" height="1"></td></tr></table><table border="0" cellspacing="0" cellpadding="0"><tr><td rowspan="5"></td> <td colspan="3" height="12"></td> <td rowspan="5"></td></tr><tr><td colspan="3"></td></tr><tr><td></td> <td><font face="Courier New, Courier, Mono New, Courier, Mono" size="2">isVoter聽=聽(age聽>=聽18);<br />isConstituent聽=聽(district聽==聽23);<br />isElector聽=聽isVoter聽&&聽isConstituent;</font></td><td></td></tr><tr><td colspan="3"></td></tr><tr><td colspan="3" height="1"></td></tr></table><table border="0" cellspacing="0" cellpadding="0"><tr><td rowspan="5"></td> <td colspan="3" height="12"></td> <td rowspan="5"></td></tr><tr><td colspan="3"></td></tr><tr><td></td> <td><font face="Times New Roman, Times, Serif" size="3">The two tables below summarize the results of applying && and 娄娄 to a pair of logical expressions (represented here by Boolean variables </font><font face="Courier New, Courier, Mono New, Courier, Mono" size="3">x</font><font face="Times New Roman, Times, Serif" size="3"> and </font><font face="Courier New, Courier, Mono New, Courier, Mono" size="3">y</font><font face="Times New Roman, Times, Serif" size="3">).</font></td><td></td></tr><tr><td colspan="3"></td></tr><tr><td colspan="3" height="1"></td></tr></table><table cellpadding="0" cellspacing="0" border="0" width="100%"><tr><td height="12"></td></tr><tr><td><table cellspacing="0" width="288" cellpadding="7"><tr><td valign="top"><font face="Times New Roman, Times, Serif" size="2"><i>Value of x</i></font></td><td valign="top"><font face="Times New Roman, Times, Serif" size="2"><i>Value of y:</i></font></td><td valign="top"><font face="Times New Roman, Times, Serif" size="2"><i>Value of x && y</i></font></td></tr><tr><td valign="top"><font face="Times New Roman, Times, Serif" size="2">TRUE</font></td><td valign="top"><table border="0" cellspacing="0" cellpadding="0" width="100%"><tr><td align="center"><font face="Times New Roman, Times, Serif" size="2">TRUE</font></td></tr></table></td><td valign="top"><table border="0" cellspacing="0" cellpadding="0" width="100%"><tr><td align="center"><font face="Times New Roman, Times, Serif" size="2">TRUE</font></td></tr></table></td></tr><tr><td valign="top"><font face="Times New Roman, Times, Serif" size="2">TRUE</font></td><td valign="top"><table border="0" cellspacing="0" cellpadding="0" width="100%"><tr><td align="center"><font face="Times New Roman, Times, Serif" size="2">FALSE</font></td></tr></table></td><td valign="top"><table border="0" cellspacing="0" cellpadding="0" width="100%"><tr><td align="center"><font face="Times New Roman, Times, Serif" size="2">FALSE</font></td></tr></table></td></tr><tr><td valign="top"><font face="Times New Roman, Times, Serif" size="2">FALSE</font></td><td valign="top"><table border="0" cellspacing="0" cellpadding="0" width="100%"><tr><td align="center"><font face="Times New Roman, Times, Serif" size="2">TRUE</font></td></tr></table></td><td valign="top"><table border="0" cellspacing="0" cellpadding="0" width="100%"><tr><td align="center"><font face="Times New Roman, Times, Serif" size="2">FALSE</font></td></tr></table></td></tr><tr><td valign="top"><font face="Times New Roman, Times, Serif" size="2">FALSE</font></td><td valign="top"><table border="0" cellspacing="0" cellpadding="0" width="100%"><tr><td align="center"><font face="Times New Roman, Times, Serif" size="2">FALSE</font></td></tr></table></td><td valign="top"><table border="0" cellspacing="0" cellpadding="0" width="100%"><tr><td align="center"><font face="Times New Roman, Times, Serif" size="2">FALSE</font></td></tr></table></td></tr></table></td></tr></table><br /><table border="0" cellspacing="0" cellpadding="0" width="100%"><tr><td rowspan="5"><img src="3e26ecb1b6ac508ae10a0e39d2fb98b2.gif" border="0" width="24" height="1" alt="3e26ecb1b6ac508ae10a0e39d2fb98b2.gif" /></td> <td colspan="3" height="12"></td> <td rowspan="5"><img src="3e26ecb1b6ac508ae10a0e39d2fb98b2.gif" border="0" width="24" height="1" alt="3e26ecb1b6ac508ae10a0e39d2fb98b2.gif" /></td></tr><tr><td colspan="3" height="1"><table cellpadding="0" cellspacing="0" border="0"><tr><td></td></tr></table></td></tr><tr><td></td> <td><font face="Times New Roman, Times, Serif" size="2">*In Boolean algebra, the pattern is formalized by a theorem called <i>DeMorgan's Law.</i></font></td><td></td></tr><tr><td colspan="3"></td></tr><tr><td colspan="3" height="1"></td></tr></table></td></tr></table><p><font size="0"></font></p>聽 </td> </tr> <tr> <td align="left" width="30%" style="background: #EEF3E2"><a 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