Adders

<div dir="ltr" style="text-align: left;" trbidi="on"> <div> <div style="text-align: justify;"> <span style="font-family: inherit;">In electronics, an <b>adder</b> or <b>summer</b> is a digital circuit that performs addition of numbers. In many computers and other kinds of processors, adders are used not only in the arithmetic logic unit(s), but also in other parts of the processor, where they are used to calculate addresses, table indices, and similar operations.</span></div> <div style="text-align: justify;"> <span style="font-family: inherit;"><br /></span></div> <div style="text-align: justify;"> <span style="font-family: inherit;">Although adders can be constructed for many numerical representations, such as binary-coded decimal or excess-3, the most common adders operate on binary numbers. In cases where two's complement or ones' complement is being used to represent negative numbers, it is trivial to modify an adder into an adder–subtractor. Other signed number representations require a more complex adder.</span></div> </div> <div style="text-align: justify;"> <span style="font-family: inherit;"><br /></span></div> <hr /> <div style="text-align: left;"> <b><span style="color: red;">HALF ADDER (HA)</span></b></div> <div style="text-align: justify;"> <br /> The half adder adds two input bits and generates a carry and sum, which are the two outputs of a half adder. The input variables of a half adder are called the augend and addend bits. The output variables are the sum and carry. The <b><u>truth table</u></b> for the half adder is;</div> <br /> <center> <table border="1" cellpadding="0" cellspacing="0" class="MsoTableGrid" style="border-collapse: collapse; border: none; mso-border-alt: solid black .5pt; mso-border-themecolor: text1; mso-padding-alt: 0in 5.4pt 0in 5.4pt;"> <tbody> <tr style="height: 22.0pt; mso-yfti-firstrow: yes; mso-yfti-irow: 0;"> <td colspan="2" style="border: solid black 1.0pt; height: 22.0pt; mso-border-alt: solid black .5pt; mso-border-themecolor: text1; mso-border-themecolor: text1; padding: 0in 5.4pt 0in 5.4pt; width: 95.4pt;" valign="top" width="127"><div align="center" class="MsoNormal" style="margin-bottom: 0.0001pt; text-align: center;"> <b><span style="font-family: "Times New Roman","serif";">INPUT <o:p></o:p></span></b></div> </td> <td colspan="2" style="border-left: none; border: solid black 1.0pt; height: 22.0pt; mso-border-alt: solid black .5pt; mso-border-left-alt: solid black .5pt; mso-border-left-themecolor: text1; mso-border-themecolor: text1; mso-border-themecolor: text1; padding: 0in 5.4pt 0in 5.4pt; width: 103.5pt;" valign="top" width="138"><div align="center" class="MsoNormal" style="margin-bottom: 0.0001pt; text-align: center;"> <b><span style="font-family: "Times New Roman","serif";">OUTPUT<o:p></o:p></span></b></div> </td> </tr> <tr style="height: 22.0pt; mso-yfti-irow: 1;"> <td style="border-top: none; border: solid black 1.0pt; height: 22.0pt; mso-border-alt: solid black .5pt; mso-border-themecolor: text1; mso-border-themecolor: text1; mso-border-top-alt: solid black .5pt; mso-border-top-themecolor: text1; padding: 0in 5.4pt 0in 5.4pt; width: 46.5pt;" valign="top" width="62"><div align="center" class="MsoNormal" style="margin-bottom: 0.0001pt; text-align: center;"> <b><span style="font-family: "Times New Roman","serif";">A<o:p></o:p></span></b></div> </td> <td style="border-bottom: solid black 1.0pt; border-left: none; border-right: solid black 1.0pt; border-top: none; height: 22.0pt; mso-border-alt: solid black .5pt; mso-border-bottom-themecolor: text1; mso-border-left-alt: solid black .5pt; mso-border-left-themecolor: text1; mso-border-right-themecolor: text1; mso-border-themecolor: text1; mso-border-top-alt: solid black .5pt; mso-border-top-themecolor: text1; padding: 0in 5.4pt 0in 5.4pt; width: 48.9pt;" valign="top" width="65"><div align="center" class="MsoNormal" style="margin-bottom: 0.0001pt; text-align: center;"> <b><span style="font-family: "Times New Roman","serif";">B<o:p></o:p></span></b></div> </td> <td style="border-bottom: solid black 1.0pt; border-left: none; border-right: solid black 1.0pt; border-top: none; height: 22.0pt; mso-border-alt: solid black .5pt; mso-border-bottom-themecolor: text1; mso-border-left-alt: solid black .5pt; mso-border-left-themecolor: text1; mso-border-right-themecolor: text1; mso-border-themecolor: text1; mso-border-top-alt: solid black .5pt; mso-border-top-themecolor: text1; padding: 0in 5.4pt 0in 5.4pt; width: 49.5pt;" valign="top" width="66"><div align="center" class="MsoNormal" style="margin-bottom: 0.0001pt; text-align: center;"> <b><span style="font-family: "Times New Roman","serif";">SUM<o:p></o:p></span></b></div> </td> <td style="border-bottom: solid black 1.0pt; border-left: none; border-right: solid black 1.0pt; border-top: none; height: 22.0pt; mso-border-alt: solid black .5pt; mso-border-bottom-themecolor: text1; mso-border-left-alt: solid black .5pt; mso-border-left-themecolor: text1; mso-border-right-themecolor: text1; mso-border-themecolor: text1; mso-border-top-alt: solid black .5pt; mso-border-top-themecolor: text1; padding: 0in 5.4pt 0in 5.4pt; width: .75in;" valign="top" width="72"><div align="center" class="MsoNormal" style="margin-bottom: 0.0001pt; text-align: center;"> <b><span style="font-family: "Times New Roman","serif";">CARRY<o:p></o:p></span></b></div> </td> </tr> <tr style="height: 22.0pt; mso-yfti-irow: 2;"> <td style="border-top: none; border: solid black 1.0pt; height: 22.0pt; mso-border-alt: solid black .5pt; mso-border-themecolor: text1; mso-border-themecolor: text1; mso-border-top-alt: solid black .5pt; mso-border-top-themecolor: text1; padding: 0in 5.4pt 0in 5.4pt; width: 46.5pt;" valign="top" width="62"><div align="center" class="MsoNormal" style="margin-bottom: 0.0001pt; text-align: center;"> 0<o:p></o:p></div> </td> <td style="border-bottom: solid black 1.0pt; border-left: none; border-right: solid black 1.0pt; border-top: none; height: 22.0pt; mso-border-alt: solid black .5pt; mso-border-bottom-themecolor: text1; mso-border-left-alt: solid black .5pt; mso-border-left-themecolor: text1; mso-border-right-themecolor: text1; mso-border-themecolor: text1; mso-border-top-alt: solid black .5pt; mso-border-top-themecolor: text1; padding: 0in 5.4pt 0in 5.4pt; width: 48.9pt;" valign="top" width="65"><div align="center" class="MsoNormal" style="margin-bottom: 0.0001pt; text-align: center;"> 0<o:p></o:p></div> </td> <td style="border-bottom: solid black 1.0pt; border-left: none; border-right: solid black 1.0pt; border-top: none; height: 22.0pt; mso-border-alt: solid black .5pt; mso-border-bottom-themecolor: text1; mso-border-left-alt: solid black .5pt; mso-border-left-themecolor: text1; mso-border-right-themecolor: text1; mso-border-themecolor: text1; mso-border-top-alt: solid black .5pt; mso-border-top-themecolor: text1; padding: 0in 5.4pt 0in 5.4pt; width: 49.5pt;" valign="top" width="66"><div align="center" class="MsoNormal" style="margin-bottom: 0.0001pt; text-align: center;"> 0<o:p></o:p></div> </td> <td style="border-bottom: solid black 1.0pt; border-left: none; border-right: solid black 1.0pt; border-top: none; height: 22.0pt; mso-border-alt: solid black .5pt; mso-border-bottom-themecolor: text1; mso-border-left-alt: solid black .5pt; mso-border-left-themecolor: text1; mso-border-right-themecolor: text1; mso-border-themecolor: text1; mso-border-top-alt: solid black .5pt; mso-border-top-themecolor: text1; padding: 0in 5.4pt 0in 5.4pt; width: .75in;" valign="top" width="72"><div align="center" class="MsoNormal" style="margin-bottom: 0.0001pt; text-align: center;"> 0<o:p></o:p></div> </td> </tr> <tr style="height: 17.5pt; mso-yfti-irow: 3;"> <td style="border-top: none; border: solid black 1.0pt; height: 17.5pt; mso-border-alt: solid black .5pt; mso-border-themecolor: text1; mso-border-themecolor: text1; mso-border-top-alt: solid black .5pt; mso-border-top-themecolor: text1; padding: 0in 5.4pt 0in 5.4pt; width: 46.5pt;" valign="top" width="62"><div align="center" class="MsoNormal" style="margin-bottom: 0.0001pt; text-align: center;"> 0<o:p></o:p></div> </td> <td style="border-bottom: solid black 1.0pt; border-left: none; border-right: solid black 1.0pt; border-top: none; height: 17.5pt; mso-border-alt: solid black .5pt; mso-border-bottom-themecolor: text1; mso-border-left-alt: solid black .5pt; mso-border-left-themecolor: text1; mso-border-right-themecolor: text1; mso-border-themecolor: text1; mso-border-top-alt: solid black .5pt; mso-border-top-themecolor: text1; padding: 0in 5.4pt 0in 5.4pt; width: 48.9pt;" valign="top" width="65"><div align="center" class="MsoNormal" style="margin-bottom: 0.0001pt; text-align: center;"> 1<o:p></o:p></div> </td> <td style="border-bottom: solid black 1.0pt; border-left: none; border-right: solid black 1.0pt; border-top: none; height: 17.5pt; mso-border-alt: solid black .5pt; mso-border-bottom-themecolor: text1; mso-border-left-alt: solid black .5pt; mso-border-left-themecolor: text1; mso-border-right-themecolor: text1; mso-border-themecolor: text1; mso-border-top-alt: solid black .5pt; mso-border-top-themecolor: text1; padding: 0in 5.4pt 0in 5.4pt; width: 49.5pt;" valign="top" width="66"><div align="center" class="MsoNormal" style="margin-bottom: 0.0001pt; text-align: center;"> 1<o:p></o:p></div> </td> <td style="border-bottom: solid black 1.0pt; border-left: none; border-right: solid black 1.0pt; border-top: none; height: 17.5pt; mso-border-alt: solid black .5pt; mso-border-bottom-themecolor: text1; mso-border-left-alt: solid black .5pt; mso-border-left-themecolor: text1; mso-border-right-themecolor: text1; mso-border-themecolor: text1; mso-border-top-alt: solid black .5pt; mso-border-top-themecolor: text1; padding: 0in 5.4pt 0in 5.4pt; width: .75in;" valign="top" width="72"><div align="center" class="MsoNormal" style="margin-bottom: 0.0001pt; text-align: center;"> 0<o:p></o:p></div> </td> </tr> <tr style="height: 17.5pt; mso-yfti-irow: 4;"> <td style="border-top: none; border: solid black 1.0pt; height: 17.5pt; mso-border-alt: solid black .5pt; mso-border-themecolor: text1; mso-border-themecolor: text1; mso-border-top-alt: solid black .5pt; mso-border-top-themecolor: text1; padding: 0in 5.4pt 0in 5.4pt; width: 46.5pt;" valign="top" width="62"><div align="center" class="MsoNormal" style="margin-bottom: 0.0001pt; text-align: center;"> 1<o:p></o:p></div> </td> <td style="border-bottom: solid black 1.0pt; border-left: none; border-right: solid black 1.0pt; border-top: none; height: 17.5pt; mso-border-alt: solid black .5pt; mso-border-bottom-themecolor: text1; mso-border-left-alt: solid black .5pt; mso-border-left-themecolor: text1; mso-border-right-themecolor: text1; mso-border-themecolor: text1; mso-border-top-alt: solid black .5pt; mso-border-top-themecolor: text1; padding: 0in 5.4pt 0in 5.4pt; width: 48.9pt;" valign="top" width="65"><div align="center" class="MsoNormal" style="margin-bottom: 0.0001pt; text-align: center;"> 0<o:p></o:p></div> </td> <td style="border-bottom: solid black 1.0pt; border-left: none; border-right: solid black 1.0pt; border-top: none; height: 17.5pt; mso-border-alt: solid black .5pt; mso-border-bottom-themecolor: text1; mso-border-left-alt: solid black .5pt; mso-border-left-themecolor: text1; mso-border-right-themecolor: text1; mso-border-themecolor: text1; mso-border-top-alt: solid black .5pt; mso-border-top-themecolor: text1; padding: 0in 5.4pt 0in 5.4pt; width: 49.5pt;" valign="top" width="66"><div align="center" class="MsoNormal" style="margin-bottom: 0.0001pt; text-align: center;"> 1<o:p></o:p></div> </td> <td style="border-bottom: solid black 1.0pt; border-left: none; border-right: solid black 1.0pt; border-top: none; height: 17.5pt; mso-border-alt: solid black .5pt; mso-border-bottom-themecolor: text1; mso-border-left-alt: solid black .5pt; mso-border-left-themecolor: text1; mso-border-right-themecolor: text1; mso-border-themecolor: text1; mso-border-top-alt: solid black .5pt; mso-border-top-themecolor: text1; padding: 0in 5.4pt 0in 5.4pt; width: .75in;" valign="top" width="72"><div align="center" class="MsoNormal" style="margin-bottom: 0.0001pt; text-align: center;"> 0<o:p></o:p></div> </td> </tr> <tr style="height: 17.5pt; mso-yfti-irow: 5; mso-yfti-lastrow: yes;"> <td style="border-top: none; border: solid black 1.0pt; height: 17.5pt; mso-border-alt: solid black .5pt; mso-border-themecolor: text1; mso-border-themecolor: text1; mso-border-top-alt: solid black .5pt; mso-border-top-themecolor: text1; padding: 0in 5.4pt 0in 5.4pt; width: 46.5pt;" valign="top" width="62"><div align="center" class="MsoNormal" style="margin-bottom: 0.0001pt; text-align: center;"> 1<o:p></o:p></div> </td> <td style="border-bottom: solid black 1.0pt; border-left: none; border-right: solid black 1.0pt; border-top: none; height: 17.5pt; mso-border-alt: solid black .5pt; mso-border-bottom-themecolor: text1; mso-border-left-alt: solid black .5pt; mso-border-left-themecolor: text1; mso-border-right-themecolor: text1; mso-border-themecolor: text1; mso-border-top-alt: solid black .5pt; mso-border-top-themecolor: text1; padding: 0in 5.4pt 0in 5.4pt; width: 48.9pt;" valign="top" width="65"><div align="center" class="MsoNormal" style="margin-bottom: 0.0001pt; text-align: center;"> 1<o:p></o:p></div> </td> <td style="border-bottom: solid black 1.0pt; border-left: none; border-right: solid black 1.0pt; border-top: none; height: 17.5pt; mso-border-alt: solid black .5pt; mso-border-bottom-themecolor: text1; mso-border-left-alt: solid black .5pt; mso-border-left-themecolor: text1; mso-border-right-themecolor: text1; mso-border-themecolor: text1; mso-border-top-alt: solid black .5pt; mso-border-top-themecolor: text1; padding: 0in 5.4pt 0in 5.4pt; width: 49.5pt;" valign="top" width="66"><div align="center" class="MsoNormal" style="margin-bottom: 0.0001pt; text-align: center;"> 0<o:p></o:p></div> </td> <td style="border-bottom: solid black 1.0pt; border-left: none; border-right: solid black 1.0pt; border-top: none; height: 17.5pt; mso-border-alt: solid black .5pt; mso-border-bottom-themecolor: text1; mso-border-left-alt: solid black .5pt; mso-border-left-themecolor: text1; mso-border-right-themecolor: text1; mso-border-themecolor: text1; mso-border-top-alt: solid black .5pt; mso-border-top-themecolor: text1; padding: 0in 5.4pt 0in 5.4pt; width: .75in;" valign="top" width="72"><div align="center" class="MsoNormal" style="margin-bottom: 0.0001pt; text-align: center;"> 1<o:p></o:p></div> </td> </tr> </tbody></table> </center> <div> <br /> From above truth table we can write <b>Boolean Expression</b> as;<br /> <div class="separator" style="clear: both; text-align: center;"> </div> <div class="separator" style="clear: both; text-align: center;"> </div> <div class="separator" style="clear: both; text-align: center;"> </div> <div class="separator" style="clear: both; text-align: center;"> <img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj47nGUglbXPPpFhyphenhyphenHRkb4yxtY5WY1_vbj7DcE9Q6GZ425s6a6DHm9df1RKI5juc9xMPf1Hx7oMdHXos6Qzut2vfmLg8mTQuciaNLX4KUaPyYuOOnWvTi687-EmntUmGtWPjKABDmGhdepw/s1600/Half+Adder.jpg" /></div> <div class="separator" style="clear: both; text-align: justify;"> S = Sum;  C = Carry</div> <div class="separator" style="clear: both; text-align: justify;"> <br /></div> <div class="separator" style="clear: both; text-align: justify;"> <b>Realization of above <u>Boolean Expressions</u> using gates;</b></div> <div class="separator" style="clear: both; text-align: center;"> <img height="235" src="http://sub.allaboutcircuits.com/images/04454.png" width="400" /></div> <div class="separator" style="clear: both; text-align: center;"> <b><i>Fig. Half-Adder using XOR & AND Gates</i></b></div> <br /> <hr /> <div style="text-align: left;"> <b><span style="color: red;">FULL ADDER (FA)</span></b></div> </div> <div class="separator" style="clear: both; text-align: center;"> </div> <div style="text-align: justify;"> <br /> This circuit  can add two binary numbers, accept a carry and yield a carry. A one-bit full adder adds three one-bit numbers, often written as A, B, and Cin; A and B are the operands, and Cin is a bit carried in from the previous less significant stage.<br /> <br /> <div class="separator" style="clear: both; text-align: center;"> <img border="0" src="https://encrypted-tbn2.gstatic.com/images?q=tbn:ANd9GcQ7dKVKegt6kuJAArFbwsnlQRNsui3XI3BnxKjssbM6bdVO8eH9Cw" /></div> <div class="separator" style="clear: both; text-align: center;"> <i><b>Fig. 1-bit Full Adder</b></i></div> </div> <div style="text-align: justify;"> <br /> Such a circuit can easily be visualized by means of two half adders (HA) and OR gate as shown in figure. A full adder can be constructed from two half adders by connecting A and B to the input of one half adder, connecting the sum from that to an input to the second adder, connecting Ci to the other input. And connect the carry of both adder to OR gate. The outputs are S and Cout.</div> <div style="text-align: center;"> <div class="separator" style="clear: both; text-align: center;"> <img src="http://doyle.wcdsb.ca/ICE3MI/Logic%20Gates/images/full%20adder%20block%20diagram.gif" /></div> <div class="separator" style="clear: both; text-align: center;"> <b><i>Fig.  Full-Adder using two Half-Adder and OR gate</i></b></div> <div class="separator" style="clear: both; text-align: justify;"> <br /></div> </div> <div style="text-align: justify;"> <b><u>Truth Table</u></b> for full adder is shown below;<br /> <div style="text-align: center;"> <br /></div> <center> <table border="1" cellpadding="0" cellspacing="0" class="MsoTableGrid" style="border-collapse: collapse; border: none; mso-border-alt: solid black .5pt; mso-border-themecolor: text1; mso-padding-alt: 0in 5.4pt 0in 5.4pt; mso-table-layout-alt: fixed;"> <tbody> <tr style="height: 22.0pt; mso-yfti-firstrow: yes; mso-yfti-irow: 0;"> <td colspan="3" style="border: solid black 1.0pt; height: 22.0pt; mso-border-alt: solid black .5pt; mso-border-themecolor: text1; mso-border-themecolor: text1; padding: 0in 5.4pt 0in 5.4pt; width: 95.4pt;" valign="top" width="127"><div align="center" class="MsoNormal" style="margin-bottom: 0.0001pt; text-align: center;"> <b><span style="font-family: "Times New Roman","serif";">INPUT <o:p></o:p></span></b></div> </td> <td colspan="2" style="border-left: none; border: solid black 1.0pt; height: 22.0pt; mso-border-alt: solid black .5pt; mso-border-left-alt: solid black .5pt; mso-border-left-themecolor: text1; mso-border-themecolor: text1; mso-border-themecolor: text1; padding: 0in 5.4pt 0in 5.4pt; width: 63.0pt;" valign="top" width="84"><div align="center" class="MsoNormal" style="margin-bottom: 0.0001pt; text-align: center;"> <b><span style="font-family: "Times New Roman","serif";">OUTPUT<o:p></o:p></span></b></div> </td> </tr> <tr style="height: 22.0pt; mso-yfti-irow: 1;"> <td style="border-top: none; border: solid black 1.0pt; height: 22.0pt; mso-border-alt: solid black .5pt; mso-border-themecolor: text1; mso-border-themecolor: text1; mso-border-top-alt: solid black .5pt; mso-border-top-themecolor: text1; padding: 0in 5.4pt 0in 5.4pt; width: .45in;" valign="top" width="43"><div align="center" class="MsoNormal" style="margin-bottom: 0.0001pt; text-align: center;"> <b><span style="font-family: "Times New Roman","serif";">A<o:p></o:p></span></b></div> </td> <td style="border-bottom: solid black 1.0pt; border-left: none; border-right: solid black 1.0pt; border-top: none; height: 22.0pt; mso-border-alt: solid black .5pt; mso-border-bottom-themecolor: text1; mso-border-left-alt: solid black .5pt; mso-border-left-themecolor: text1; mso-border-right-themecolor: text1; mso-border-themecolor: text1; mso-border-top-alt: solid black .5pt; mso-border-top-themecolor: text1; padding: 0in 5.4pt 0in 5.4pt; width: 31.5pt;" valign="top" width="42"><div align="center" class="MsoNormal" style="margin-bottom: 0.0001pt; text-align: center;"> <b><span style="font-family: "Times New Roman","serif";">B<o:p></o:p></span></b></div> </td> <td style="border-bottom: solid black 1.0pt; border-left: none; border-right: solid black 1.0pt; border-top: none; height: 22.0pt; mso-border-alt: solid black .5pt; mso-border-bottom-themecolor: text1; mso-border-left-alt: solid black .5pt; mso-border-left-themecolor: text1; mso-border-right-themecolor: text1; mso-border-themecolor: text1; mso-border-top-alt: solid black .5pt; mso-border-top-themecolor: text1; padding: 0in 5.4pt 0in 5.4pt; width: 31.5pt;" valign="top" width="42"><div align="center" class="MsoNormal" style="margin-bottom: 0.0001pt; text-align: center;"> <b><span style="font-family: "Times New Roman","serif";">Ci<o:p></o:p></span></b></div> </td> <td style="border-bottom: solid black 1.0pt; border-left: none; border-right: solid black 1.0pt; border-top: none; height: 22.0pt; mso-border-alt: solid black .5pt; mso-border-bottom-themecolor: text1; mso-border-left-alt: solid black .5pt; mso-border-left-themecolor: text1; mso-border-right-themecolor: text1; mso-border-themecolor: text1; mso-border-top-alt: solid black .5pt; mso-border-top-themecolor: text1; padding: 0in 5.4pt 0in 5.4pt; width: 31.5pt;" valign="top" width="42"><div align="center" class="MsoNormal" style="margin-bottom: 0.0001pt; text-align: center;"> <b><span style="font-family: "Times New Roman","serif";">S<o:p></o:p></span></b></div> </td> <td style="border-bottom: solid black 1.0pt; border-left: none; border-right: solid black 1.0pt; border-top: none; height: 22.0pt; mso-border-alt: solid black .5pt; mso-border-bottom-themecolor: text1; mso-border-left-alt: solid black .5pt; mso-border-left-themecolor: text1; mso-border-right-themecolor: text1; mso-border-themecolor: text1; mso-border-top-alt: solid black .5pt; mso-border-top-themecolor: text1; padding: 0in 5.4pt 0in 5.4pt; width: 31.5pt;" valign="top" width="42"><div align="center" class="MsoNormal" style="margin-bottom: 0.0001pt; text-align: center;"> <b><span style="font-family: "Times New Roman","serif";">Co<o:p></o:p></span></b></div> </td> </tr> <tr style="height: 22.0pt; mso-yfti-irow: 2;"> <td style="border-top: none; border: solid black 1.0pt; height: 22.0pt; mso-border-alt: solid black .5pt; mso-border-themecolor: text1; mso-border-themecolor: text1; mso-border-top-alt: solid black .5pt; mso-border-top-themecolor: text1; padding: 0in 5.4pt 0in 5.4pt; width: .45in;" valign="top" width="43"><div align="center" class="MsoNormal" style="margin-bottom: 0.0001pt; text-align: center;"> 0<o:p></o:p></div> </td> <td style="border-bottom: solid black 1.0pt; border-left: none; border-right: solid black 1.0pt; border-top: none; height: 22.0pt; mso-border-alt: solid black .5pt; mso-border-bottom-themecolor: text1; mso-border-left-alt: solid black .5pt; mso-border-left-themecolor: text1; mso-border-right-themecolor: text1; mso-border-themecolor: text1; mso-border-top-alt: solid black .5pt; mso-border-top-themecolor: text1; padding: 0in 5.4pt 0in 5.4pt; width: 31.5pt;" valign="top" width="42"><div align="center" class="MsoNormal" style="margin-bottom: 0.0001pt; text-align: center;"> 0<o:p></o:p></div> </td> <td style="border-bottom: solid black 1.0pt; border-left: none; border-right: solid black 1.0pt; border-top: none; height: 22.0pt; mso-border-alt: solid black .5pt; mso-border-bottom-themecolor: text1; mso-border-left-alt: solid black .5pt; mso-border-left-themecolor: text1; mso-border-right-themecolor: text1; mso-border-themecolor: text1; mso-border-top-alt: solid black .5pt; mso-border-top-themecolor: text1; padding: 0in 5.4pt 0in 5.4pt; width: 31.5pt;" valign="top" width="42"><div align="center" class="MsoNormal" style="margin-bottom: 0.0001pt; text-align: center;"> 0<o:p></o:p></div> </td> <td style="border-bottom: solid black 1.0pt; border-left: none; border-right: solid black 1.0pt; border-top: none; height: 22.0pt; mso-border-alt: solid black .5pt; mso-border-bottom-themecolor: text1; mso-border-left-alt: solid black .5pt; mso-border-left-themecolor: text1; mso-border-right-themecolor: text1; mso-border-themecolor: text1; mso-border-top-alt: solid black .5pt; mso-border-top-themecolor: text1; padding: 0in 5.4pt 0in 5.4pt; width: 31.5pt;" valign="top" width="42"><div align="center" class="MsoNormal" style="margin-bottom: 0.0001pt; text-align: center;"> 0<o:p></o:p></div> </td> <td style="border-bottom: solid black 1.0pt; border-left: none; border-right: solid black 1.0pt; border-top: none; height: 22.0pt; mso-border-alt: solid black .5pt; mso-border-bottom-themecolor: text1; mso-border-left-alt: solid black .5pt; mso-border-left-themecolor: text1; mso-border-right-themecolor: text1; mso-border-themecolor: text1; mso-border-top-alt: solid black .5pt; mso-border-top-themecolor: text1; padding: 0in 5.4pt 0in 5.4pt; width: 31.5pt;" valign="top" width="42"><div align="center" class="MsoNormal" style="margin-bottom: 0.0001pt; text-align: center;"> 0<o:p></o:p></div> </td> </tr> <tr style="height: 17.5pt; mso-yfti-irow: 3;"> <td style="border-top: none; border: solid black 1.0pt; height: 17.5pt; mso-border-alt: solid black .5pt; mso-border-themecolor: text1; mso-border-themecolor: text1; mso-border-top-alt: solid black .5pt; mso-border-top-themecolor: text1; padding: 0in 5.4pt 0in 5.4pt; width: .45in;" valign="top" width="43"><div align="center" class="MsoNormal" style="margin-bottom: 0.0001pt; text-align: center;"> 0<o:p></o:p></div> </td> <td style="border-bottom: solid black 1.0pt; border-left: none; border-right: solid black 1.0pt; border-top: none; height: 17.5pt; mso-border-alt: solid black .5pt; mso-border-bottom-themecolor: text1; mso-border-left-alt: solid black .5pt; mso-border-left-themecolor: text1; mso-border-right-themecolor: text1; mso-border-themecolor: text1; mso-border-top-alt: solid black .5pt; mso-border-top-themecolor: text1; padding: 0in 5.4pt 0in 5.4pt; width: 31.5pt;" valign="top" width="42"><div align="center" class="MsoNormal" style="margin-bottom: 0.0001pt; text-align: center;"> 0<o:p></o:p></div> </td> <td style="border-bottom: solid black 1.0pt; border-left: none; border-right: solid black 1.0pt; border-top: none; height: 17.5pt; mso-border-alt: solid black .5pt; mso-border-bottom-themecolor: text1; mso-border-left-alt: solid black .5pt; mso-border-left-themecolor: text1; mso-border-right-themecolor: text1; mso-border-themecolor: text1; mso-border-top-alt: solid black .5pt; mso-border-top-themecolor: text1; padding: 0in 5.4pt 0in 5.4pt; width: 31.5pt;" valign="top" width="42"><div align="center" class="MsoNormal" style="margin-bottom: 0.0001pt; text-align: center;"> 1<o:p></o:p></div> </td> <td style="border-bottom: solid black 1.0pt; border-left: none; border-right: solid black 1.0pt; border-top: none; height: 17.5pt; mso-border-alt: solid black .5pt; mso-border-bottom-themecolor: text1; mso-border-left-alt: solid black .5pt; mso-border-left-themecolor: text1; mso-border-right-themecolor: text1; mso-border-themecolor: text1; mso-border-top-alt: solid black .5pt; mso-border-top-themecolor: text1; padding: 0in 5.4pt 0in 5.4pt; width: 31.5pt;" valign="top" width="42"><div align="center" class="MsoNormal" style="margin-bottom: 0.0001pt; text-align: center;"> 0<o:p></o:p></div> </td> <td style="border-bottom: solid black 1.0pt; border-left: none; border-right: solid black 1.0pt; border-top: none; height: 17.5pt; mso-border-alt: solid black .5pt; mso-border-bottom-themecolor: text1; mso-border-left-alt: solid black .5pt; mso-border-left-themecolor: text1; mso-border-right-themecolor: text1; mso-border-themecolor: text1; mso-border-top-alt: solid black .5pt; mso-border-top-themecolor: text1; padding: 0in 5.4pt 0in 5.4pt; width: 31.5pt;" valign="top" width="42"><div align="center" class="MsoNormal" style="margin-bottom: 0.0001pt; text-align: center;"> 1<o:p></o:p></div> </td> </tr> <tr style="height: 17.5pt; mso-yfti-irow: 4;"> <td style="border-top: none; border: solid black 1.0pt; height: 17.5pt; mso-border-alt: solid black .5pt; mso-border-themecolor: text1; mso-border-themecolor: text1; mso-border-top-alt: solid black .5pt; mso-border-top-themecolor: text1; padding: 0in 5.4pt 0in 5.4pt; width: .45in;" valign="top" width="43"><div align="center" class="MsoNormal" style="margin-bottom: 0.0001pt; text-align: center;"> 0<o:p></o:p></div> </td> <td style="border-bottom: solid black 1.0pt; border-left: none; border-right: solid black 1.0pt; border-top: none; height: 17.5pt; mso-border-alt: solid black .5pt; mso-border-bottom-themecolor: text1; mso-border-left-alt: solid black .5pt; mso-border-left-themecolor: text1; mso-border-right-themecolor: text1; mso-border-themecolor: text1; mso-border-top-alt: solid black .5pt; mso-border-top-themecolor: text1; padding: 0in 5.4pt 0in 5.4pt; width: 31.5pt;" valign="top" width="42"><div align="center" class="MsoNormal" style="margin-bottom: 0.0001pt; text-align: center;"> 1<o:p></o:p></div> </td> <td style="border-bottom: solid black 1.0pt; border-left: none; border-right: solid black 1.0pt; border-top: none; height: 17.5pt; mso-border-alt: solid black .5pt; mso-border-bottom-themecolor: text1; mso-border-left-alt: solid black .5pt; mso-border-left-themecolor: text1; mso-border-right-themecolor: text1; mso-border-themecolor: text1; mso-border-top-alt: solid black .5pt; mso-border-top-themecolor: text1; padding: 0in 5.4pt 0in 5.4pt; width: 31.5pt;" valign="top" width="42"><div align="center" class="MsoNormal" style="margin-bottom: 0.0001pt; text-align: center;"> 0<o:p></o:p></div> </td> <td style="border-bottom: solid black 1.0pt; border-left: none; border-right: solid black 1.0pt; border-top: none; height: 17.5pt; mso-border-alt: solid black .5pt; mso-border-bottom-themecolor: text1; mso-border-left-alt: solid black .5pt; mso-border-left-themecolor: text1; mso-border-right-themecolor: text1; mso-border-themecolor: text1; mso-border-top-alt: solid black .5pt; mso-border-top-themecolor: text1; padding: 0in 5.4pt 0in 5.4pt; width: 31.5pt;" valign="top" width="42"><div align="center" class="MsoNormal" style="margin-bottom: 0.0001pt; text-align: center;"> 0<o:p></o:p></div> </td> <td style="border-bottom: solid black 1.0pt; border-left: none; border-right: solid black 1.0pt; border-top: none; height: 17.5pt; mso-border-alt: solid black .5pt; mso-border-bottom-themecolor: text1; mso-border-left-alt: solid black .5pt; mso-border-left-themecolor: text1; mso-border-right-themecolor: text1; mso-border-themecolor: text1; mso-border-top-alt: solid black .5pt; mso-border-top-themecolor: text1; padding: 0in 5.4pt 0in 5.4pt; width: 31.5pt;" valign="top" width="42"><div align="center" class="MsoNormal" style="margin-bottom: 0.0001pt; text-align: center;"> 1<o:p></o:p></div> </td> </tr> <tr style="height: 17.5pt; mso-yfti-irow: 5;"> <td style="border-top: none; border: solid black 1.0pt; height: 17.5pt; mso-border-alt: solid black .5pt; mso-border-themecolor: text1; mso-border-themecolor: text1; mso-border-top-alt: solid black .5pt; mso-border-top-themecolor: text1; padding: 0in 5.4pt 0in 5.4pt; width: .45in;" valign="top" width="43"><div align="center" class="MsoNormal" style="margin-bottom: 0.0001pt; text-align: center;"> 0<o:p></o:p></div> </td> <td style="border-bottom: solid black 1.0pt; border-left: none; border-right: solid black 1.0pt; border-top: none; height: 17.5pt; mso-border-alt: solid black .5pt; mso-border-bottom-themecolor: text1; mso-border-left-alt: solid black .5pt; mso-border-left-themecolor: text1; mso-border-right-themecolor: text1; mso-border-themecolor: text1; mso-border-top-alt: solid black .5pt; mso-border-top-themecolor: text1; padding: 0in 5.4pt 0in 5.4pt; width: 31.5pt;" valign="top" width="42"><div align="center" class="MsoNormal" style="margin-bottom: 0.0001pt; text-align: center;"> 1<o:p></o:p></div> </td> <td style="border-bottom: solid black 1.0pt; border-left: none; border-right: solid black 1.0pt; border-top: none; height: 17.5pt; mso-border-alt: solid black .5pt; mso-border-bottom-themecolor: text1; mso-border-left-alt: solid black .5pt; mso-border-left-themecolor: text1; mso-border-right-themecolor: text1; mso-border-themecolor: text1; mso-border-top-alt: solid black .5pt; mso-border-top-themecolor: text1; padding: 0in 5.4pt 0in 5.4pt; width: 31.5pt;" valign="top" width="42"><div align="center" class="MsoNormal" style="margin-bottom: 0.0001pt; text-align: center;"> 1<o:p></o:p></div> </td> <td style="border-bottom: solid black 1.0pt; border-left: none; border-right: solid black 1.0pt; border-top: none; height: 17.5pt; mso-border-alt: solid black .5pt; mso-border-bottom-themecolor: text1; mso-border-left-alt: solid black .5pt; mso-border-left-themecolor: text1; mso-border-right-themecolor: text1; mso-border-themecolor: text1; mso-border-top-alt: solid black .5pt; mso-border-top-themecolor: text1; padding: 0in 5.4pt 0in 5.4pt; width: 31.5pt;" valign="top" width="42"><div align="center" class="MsoNormal" style="margin-bottom: 0.0001pt; text-align: center;"> 1<o:p></o:p></div> </td> <td style="border-bottom: solid black 1.0pt; border-left: none; border-right: solid black 1.0pt; border-top: none; height: 17.5pt; mso-border-alt: solid black .5pt; mso-border-bottom-themecolor: text1; mso-border-left-alt: solid black .5pt; mso-border-left-themecolor: text1; mso-border-right-themecolor: text1; mso-border-themecolor: text1; mso-border-top-alt: solid black .5pt; mso-border-top-themecolor: text1; padding: 0in 5.4pt 0in 5.4pt; width: 31.5pt;" valign="top" width="42"><div align="center" class="MsoNormal" style="margin-bottom: 0.0001pt; text-align: center;"> 0<o:p></o:p></div> </td> </tr> <tr style="height: 17.5pt; mso-yfti-irow: 6;"> <td style="border-top: none; border: solid black 1.0pt; height: 17.5pt; mso-border-alt: solid black .5pt; mso-border-themecolor: text1; mso-border-themecolor: text1; mso-border-top-alt: solid black .5pt; mso-border-top-themecolor: text1; padding: 0in 5.4pt 0in 5.4pt; width: .45in;" valign="top" width="43"><div align="center" class="MsoNormal" style="margin-bottom: 0.0001pt; text-align: center;"> 1<o:p></o:p></div> </td> <td style="border-bottom: solid black 1.0pt; border-left: none; border-right: solid black 1.0pt; border-top: none; height: 17.5pt; mso-border-alt: solid black .5pt; mso-border-bottom-themecolor: text1; mso-border-left-alt: solid black .5pt; mso-border-left-themecolor: text1; mso-border-right-themecolor: text1; mso-border-themecolor: text1; mso-border-top-alt: solid black .5pt; mso-border-top-themecolor: text1; padding: 0in 5.4pt 0in 5.4pt; width: 31.5pt;" valign="top" width="42"><div align="center" class="MsoNormal" style="margin-bottom: 0.0001pt; text-align: center;"> 0<o:p></o:p></div> </td> <td style="border-bottom: solid black 1.0pt; border-left: none; border-right: solid black 1.0pt; border-top: none; height: 17.5pt; mso-border-alt: solid black .5pt; mso-border-bottom-themecolor: text1; mso-border-left-alt: solid black .5pt; mso-border-left-themecolor: text1; mso-border-right-themecolor: text1; mso-border-themecolor: text1; mso-border-top-alt: solid black .5pt; mso-border-top-themecolor: text1; padding: 0in 5.4pt 0in 5.4pt; width: 31.5pt;" valign="top" width="42"><div align="center" class="MsoNormal" style="margin-bottom: 0.0001pt; text-align: center;"> 0<o:p></o:p></div> </td> <td style="border-bottom: solid black 1.0pt; border-left: none; border-right: solid black 1.0pt; border-top: none; height: 17.5pt; mso-border-alt: solid black .5pt; mso-border-bottom-themecolor: text1; mso-border-left-alt: solid black .5pt; mso-border-left-themecolor: text1; mso-border-right-themecolor: text1; mso-border-themecolor: text1; mso-border-top-alt: solid black .5pt; mso-border-top-themecolor: text1; padding: 0in 5.4pt 0in 5.4pt; width: 31.5pt;" valign="top" width="42"><div align="center" class="MsoNormal" style="margin-bottom: 0.0001pt; text-align: center;"> 0<o:p></o:p></div> </td> <td style="border-bottom: solid black 1.0pt; border-left: none; border-right: solid black 1.0pt; border-top: none; height: 17.5pt; mso-border-alt: solid black .5pt; mso-border-bottom-themecolor: text1; mso-border-left-alt: solid black .5pt; mso-border-left-themecolor: text1; mso-border-right-themecolor: text1; mso-border-themecolor: text1; mso-border-top-alt: solid black .5pt; mso-border-top-themecolor: text1; padding: 0in 5.4pt 0in 5.4pt; width: 31.5pt;" valign="top" width="42"><div align="center" class="MsoNormal" style="margin-bottom: 0.0001pt; text-align: center;"> 1<o:p></o:p></div> </td> </tr> <tr style="height: 17.5pt; mso-yfti-irow: 7;"> <td style="border-top: none; border: solid black 1.0pt; height: 17.5pt; mso-border-alt: solid black .5pt; mso-border-themecolor: text1; mso-border-themecolor: text1; mso-border-top-alt: solid black .5pt; mso-border-top-themecolor: text1; padding: 0in 5.4pt 0in 5.4pt; width: .45in;" valign="top" width="43"><div align="center" class="MsoNormal" style="margin-bottom: 0.0001pt; text-align: center;"> 1<o:p></o:p></div> </td> <td style="border-bottom: solid black 1.0pt; border-left: none; border-right: solid black 1.0pt; border-top: none; height: 17.5pt; mso-border-alt: solid black .5pt; mso-border-bottom-themecolor: text1; mso-border-left-alt: solid black .5pt; mso-border-left-themecolor: text1; mso-border-right-themecolor: text1; mso-border-themecolor: text1; mso-border-top-alt: solid black .5pt; mso-border-top-themecolor: text1; padding: 0in 5.4pt 0in 5.4pt; width: 31.5pt;" valign="top" width="42"><div align="center" class="MsoNormal" style="margin-bottom: 0.0001pt; text-align: center;"> 0<o:p></o:p></div> </td> <td style="border-bottom: solid black 1.0pt; border-left: none; border-right: solid black 1.0pt; border-top: none; height: 17.5pt; mso-border-alt: solid black .5pt; mso-border-bottom-themecolor: text1; mso-border-left-alt: solid black .5pt; mso-border-left-themecolor: text1; mso-border-right-themecolor: text1; mso-border-themecolor: text1; mso-border-top-alt: solid black .5pt; mso-border-top-themecolor: text1; padding: 0in 5.4pt 0in 5.4pt; width: 31.5pt;" valign="top" width="42"><div align="center" class="MsoNormal" style="margin-bottom: 0.0001pt; text-align: center;"> 1<o:p></o:p></div> </td> <td style="border-bottom: solid black 1.0pt; border-left: none; border-right: solid black 1.0pt; border-top: none; height: 17.5pt; mso-border-alt: solid black .5pt; mso-border-bottom-themecolor: text1; mso-border-left-alt: solid black .5pt; mso-border-left-themecolor: text1; mso-border-right-themecolor: text1; mso-border-themecolor: text1; mso-border-top-alt: solid black .5pt; mso-border-top-themecolor: text1; padding: 0in 5.4pt 0in 5.4pt; width: 31.5pt;" valign="top" width="42"><div align="center" class="MsoNormal" style="margin-bottom: 0.0001pt; text-align: center;"> 1<o:p></o:p></div> </td> <td style="border-bottom: solid black 1.0pt; border-left: none; border-right: solid black 1.0pt; border-top: none; height: 17.5pt; mso-border-alt: solid black .5pt; mso-border-bottom-themecolor: text1; mso-border-left-alt: solid black .5pt; mso-border-left-themecolor: text1; mso-border-right-themecolor: text1; mso-border-themecolor: text1; mso-border-top-alt: solid black .5pt; mso-border-top-themecolor: text1; padding: 0in 5.4pt 0in 5.4pt; width: 31.5pt;" valign="top" width="42"><div align="center" class="MsoNormal" style="margin-bottom: 0.0001pt; text-align: center;"> 0<o:p></o:p></div> </td> </tr> <tr style="height: 17.5pt; mso-yfti-irow: 8;"> <td style="border-top: none; border: solid black 1.0pt; height: 17.5pt; mso-border-alt: solid black .5pt; mso-border-themecolor: text1; mso-border-themecolor: text1; mso-border-top-alt: solid black .5pt; mso-border-top-themecolor: text1; padding: 0in 5.4pt 0in 5.4pt; width: .45in;" valign="top" width="43"><div align="center" class="MsoNormal" style="margin-bottom: 0.0001pt; text-align: center;"> 1<o:p></o:p></div> </td> <td style="border-bottom: solid black 1.0pt; border-left: none; border-right: solid black 1.0pt; border-top: none; height: 17.5pt; mso-border-alt: solid black .5pt; mso-border-bottom-themecolor: text1; mso-border-left-alt: solid black .5pt; mso-border-left-themecolor: text1; mso-border-right-themecolor: text1; mso-border-themecolor: text1; mso-border-top-alt: solid black .5pt; mso-border-top-themecolor: text1; padding: 0in 5.4pt 0in 5.4pt; width: 31.5pt;" valign="top" width="42"><div align="center" class="MsoNormal" style="margin-bottom: 0.0001pt; text-align: center;"> 1<o:p></o:p></div> </td> <td style="border-bottom: solid black 1.0pt; border-left: none; border-right: solid black 1.0pt; border-top: none; height: 17.5pt; mso-border-alt: solid black .5pt; mso-border-bottom-themecolor: text1; mso-border-left-alt: solid black .5pt; mso-border-left-themecolor: text1; mso-border-right-themecolor: text1; mso-border-themecolor: text1; mso-border-top-alt: solid black .5pt; mso-border-top-themecolor: text1; padding: 0in 5.4pt 0in 5.4pt; width: 31.5pt;" valign="top" width="42"><div align="center" class="MsoNormal" style="margin-bottom: 0.0001pt; text-align: center;"> 0<o:p></o:p></div> </td> <td style="border-bottom: solid black 1.0pt; border-left: none; border-right: solid black 1.0pt; border-top: none; height: 17.5pt; mso-border-alt: solid black .5pt; mso-border-bottom-themecolor: text1; mso-border-left-alt: solid black .5pt; mso-border-left-themecolor: text1; mso-border-right-themecolor: text1; mso-border-themecolor: text1; mso-border-top-alt: solid black .5pt; mso-border-top-themecolor: text1; padding: 0in 5.4pt 0in 5.4pt; width: 31.5pt;" valign="top" width="42"><div align="center" class="MsoNormal" style="margin-bottom: 0.0001pt; text-align: center;"> 1<o:p></o:p></div> </td> <td style="border-bottom: solid black 1.0pt; border-left: none; border-right: solid black 1.0pt; border-top: none; height: 17.5pt; mso-border-alt: solid black .5pt; mso-border-bottom-themecolor: text1; mso-border-left-alt: solid black .5pt; mso-border-left-themecolor: text1; mso-border-right-themecolor: text1; mso-border-themecolor: text1; mso-border-top-alt: solid black .5pt; mso-border-top-themecolor: text1; padding: 0in 5.4pt 0in 5.4pt; width: 31.5pt;" valign="top" width="42"><div align="center" class="MsoNormal" style="margin-bottom: 0.0001pt; text-align: center;"> 0<o:p></o:p></div> </td> </tr> <tr style="height: 17.5pt; mso-yfti-irow: 9; mso-yfti-lastrow: yes;"> <td style="border-top: none; border: solid black 1.0pt; height: 17.5pt; mso-border-alt: solid black .5pt; mso-border-themecolor: text1; mso-border-themecolor: text1; mso-border-top-alt: solid black .5pt; mso-border-top-themecolor: text1; padding: 0in 5.4pt 0in 5.4pt; width: .45in;" valign="top" width="43"><div align="center" class="MsoNormal" style="margin-bottom: 0.0001pt; text-align: center;"> 1<o:p></o:p></div> </td> <td style="border-bottom: solid black 1.0pt; border-left: none; border-right: solid black 1.0pt; border-top: none; height: 17.5pt; mso-border-alt: solid black .5pt; mso-border-bottom-themecolor: text1; mso-border-left-alt: solid black .5pt; mso-border-left-themecolor: text1; mso-border-right-themecolor: text1; mso-border-themecolor: text1; mso-border-top-alt: solid black .5pt; mso-border-top-themecolor: text1; padding: 0in 5.4pt 0in 5.4pt; width: 31.5pt;" valign="top" width="42"><div align="center" class="MsoNormal" style="margin-bottom: 0.0001pt; text-align: center;"> 1<o:p></o:p></div> </td> <td style="border-bottom: solid black 1.0pt; border-left: none; border-right: solid black 1.0pt; border-top: none; height: 17.5pt; mso-border-alt: solid black .5pt; mso-border-bottom-themecolor: text1; mso-border-left-alt: solid black .5pt; mso-border-left-themecolor: text1; mso-border-right-themecolor: text1; mso-border-themecolor: text1; mso-border-top-alt: solid black .5pt; mso-border-top-themecolor: text1; padding: 0in 5.4pt 0in 5.4pt; width: 31.5pt;" valign="top" width="42"><div align="center" class="MsoNormal" style="margin-bottom: 0.0001pt; text-align: center;"> 1<o:p></o:p></div> </td> <td style="border-bottom: solid black 1.0pt; border-left: none; border-right: solid black 1.0pt; border-top: none; height: 17.5pt; mso-border-alt: solid black .5pt; mso-border-bottom-themecolor: text1; mso-border-left-alt: solid black .5pt; mso-border-left-themecolor: text1; mso-border-right-themecolor: text1; mso-border-themecolor: text1; mso-border-top-alt: solid black .5pt; mso-border-top-themecolor: text1; padding: 0in 5.4pt 0in 5.4pt; width: 31.5pt;" valign="top" width="42"><div align="center" class="MsoNormal" style="margin-bottom: 0.0001pt; text-align: center;"> 1<o:p></o:p></div> </td> <td style="border-bottom: solid black 1.0pt; border-left: none; border-right: solid black 1.0pt; border-top: none; height: 17.5pt; mso-border-alt: solid black .5pt; mso-border-bottom-themecolor: text1; mso-border-left-alt: solid black .5pt; mso-border-left-themecolor: text1; mso-border-right-themecolor: text1; mso-border-themecolor: text1; mso-border-top-alt: solid black .5pt; mso-border-top-themecolor: text1; padding: 0in 5.4pt 0in 5.4pt; width: 31.5pt;" valign="top" width="42"><div align="center" class="MsoNormal" style="margin-bottom: 0.0001pt; text-align: center;"> 1<o:p></o:p></div> </td></tr> </tbody></table> </center> </div> <div style="text-align: center;"> <br /></div> <div style="text-align: center;"> </div> <div style="text-align: justify;"> From above truth table we can write <b>Boolean Expression</b> as;</div> <div style="text-align: center;"> <img alt="S = A \oplus B \oplus C_{in}" src="http://upload.wikimedia.org/math/9/c/1/9c1accbfe0c7a7c9b86e258eda968f1c.png" /></div> <div style="text-align: center;"> <img alt="C_{out} = (A \cdot B) + (C_{in} \cdot (A \oplus B))" src="http://upload.wikimedia.org/math/b/d/e/bdec2f97f269fd6f4ef3857c2306c3f2.png" /></div> <div style="text-align: justify;"> <br /></div> <div style="text-align: justify;"> <b>Realization of above <u>Boolean Expression</u> using gates;</b></div> <div style="text-align: center;"> <div class="separator" style="clear: both; text-align: center;"> <img alt="Full Adder Circuit" src="http://www.circuitstoday.com/wp-content/uploads/2010/04/Full-Adder-Circuit.gif" /></div> <div class="separator" style="clear: both; text-align: center;"> <b><i>Fig. Realization of Full-Adder using Boolean Expression</i></b></div> </div> </div>

Adders

In electronics, an adder or summer is a digital circuit that performs addition of numbers. In many computers and other kinds of process...

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