SUM (Ternary Gate): Difference between revisions

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<p>
<big><b>Modulo-3 Sum</b></big>
Modulo-3 Sum
[[File:SUM_GATE.png|thumb|Sum Gate Symbol]]
</p>
[[File:BCT_SUM.png|thumb|BCT Sum Gate]]
== Uses ==
While there is a [[EOR (Ternary Gate)|Ternary XOR]] gate, it's usefulness does not fit with what the [[XOR (Binary Gate)|Binary XOR]] can do. Another gate is needed to perform ternary addition.
 
SUM does mod 3 addition, returning the remainder.
 
It is also useful for cryptography and error correction being balanced, self-negating, associative, and commutative.
 
<i>
A ⊕ B = C<br />
C ⊕ A = B<br />
A ⊕ C = B<br />
B ⊕ C = A<br />
A ⊕ 0 = A<br />
0 ⊕ A = A<br />
</i>


== Truth Tables ==
== Truth Tables ==
<div style="text-align: center; font-family: monospace; font-size: 20px;">
=== SUM ===
<table style="margin: 50px; display: inline-block; vertical-align: text-top; border-collapse: collapse;">
<div class="tt">
<tr>
<table class="tt">
<td class="tct" colspan="2" rowspan="2">SUM</td>
<tr>
<td colspan="3" class="tce">B</td>
<td class="tt_br tt_bb" colspan="2" rowspan="2">SUM</td>
</tr>
<td colspan="3" class="tce"><b>B</b></td>
<tr>
</tr>
<td class="tcb1">-</td>
<tr>
<td class="tcb2">0</td>
<td class="tt_r tt_bb">-</td>
<td class="tcb3">+</td>
<td class="tt_g tt_bb">0</td>
</tr>
<td class="tt_b tt_bb">+</td>
<tr>
</tr>
<td rowspan="3" class="tca">A</td>
<tr>
<td class="tcr1">-</td>
<td rowspan="3"><b>A</b></td>
<td class="tc3">+</td>
<td class="tt_r tt_br">-</td>
<td class="tc1">-</td>
<td class="tt_b">+</td>
<td class="tc2">0</td>
<td class="tt_r">-</td>
</tr>
<td class="tt_g">0</td>
<tr>
</tr>
<td class="tcr2">0</td>
<tr>
<td class="tc1">-</td>
<td class="tt_g tt_br">0</td>
<td class="tc2">0</td>
<td class="tt_r">-</td>
<td class="tc3">+</td>
<td class="tt_g">0</td>
</tr>
<td class="tt_b">+</td>
<tr>
</tr>
<td class="tcr3">+</td>
<tr>
<td class="tc2">0</td>
<td class="tt_b tt_br">+</td>
<td class="tc3">+</td>
<td class="tt_g">0</td>
<td class="tc1">-</td>
<td class="tt_b">+</td>
</tr>
<td class="tt_r">-</td>
</table>
</tr>
</table>
 
<table class="tt">
<tr>
<td colspan="3">SUM</td>
</tr>
<tr>
<td class="tt_bb"><b>A</b></td>
<td class="tt_bb"><b>B</b></td>
<td class="tt_bl tt_bb"><b>Y</b></td>
</tr>
<tr>
<td class="tt_r">-</td>
<td class="tt_r">-</td>
<td class="tt_bl tt_b">+</td>
</tr>
<tr>
<td class="tt_r">-</td>
<td class="tt_g">0</td>
<td class="tt_bl tt_r">-</td>
</tr>
<tr>
<td class="tt_r">-</td>
<td class="tt_b">+</td>
<td class="tt_bl tt_g">0</td>
</tr>
<tr>
<td class="tt_g">0</td>
<td class="tt_r">-</td>
<td class="tt_bl tt_r">-</td>
</tr>
<tr>
<td class="tt_g">0</td>
<td class="tt_g">0</td>
<td class="tt_bl tt_g">0</td>
</tr>
<tr>
<td class="tt_g">0</td>
<td class="tt_b">+</td>
<td class="tt_bl tt_b">+</td>
</tr>
<tr>
<td class="tt_b">+</td>
<td class="tt_r">-</td>
<td class="tt_bl tt_g">0</td>
</tr>
<tr>
<td class="tt_b">+</td>
<td class="tt_g">0</td>
<td class="tt_bl tt_b">+</td>
</tr>
<tr>
<td class="tt_b">+</td>
<td class="tt_b">+</td>
<td class="tt_bl tt_r">-</td>
</tr>
</table>
</div>
 
<hr />
 
=== NSUM ===
<div class="tt">
<table class="tt">
<tr>
<td class="tt_br tt_bb" colspan="2" rowspan="2">NSUM</td>
<td colspan="3" class="tce"><b>B</b></td>
</tr>
<tr>
<td class="tt_r tt_bb">-</td>
<td class="tt_g tt_bb">0</td>
<td class="tt_b tt_bb">+</td>
</tr>
<tr>
<td rowspan="3"><b>A</b></td>
<td class="tt_r tt_br">-</td>
<td class="tt_r">-</td>
<td class="tt_b">+</td>
<td class="tt_g">0</td>
</tr>
<tr>
<td class="tt_g tt_br">0</td>
<td class="tt_b">+</td>
<td class="tt_g">0</td>
<td class="tt_r">-</td>
</tr>
<tr>
<td class="tt_b tt_br">+</td>
<td class="tt_g">0</td>
<td class="tt_r">-</td>
<td class="tt_b">+</td>
</tr>
</table>


<table style="margin: 50px; display: inline-block; vertical-align: text-top; border-collapse: collapse;">
<table class="tt">
<tr>
<tr>
<td class="ttb">A</td>
<td colspan="3">NSUM</td>
<td class="ttb">B</td>
</tr>
<td class="ttb">C</td>
<tr>
</tr>
<td class="tt_bb"><b>A</b></td>
<tr>
<td class="tt_bb"><b>B</b></td>
<td>-</td>
<td class="tt_bl tt_bb"><b>Y</b></td>
<td>-</td>
</tr>
<td>+</td>
<tr>
</tr>
<td class="tt_r">-</td>
<tr>
<td class="tt_r">-</td>
<td>-</td>
<td class="tt_bl tt_r">-</td>
<td>0</td>
</tr>
<td>C</td>
<tr>
</tr>
<td class="tt_r">-</td>
<tr>
<td class="tt_g">0</td>
<td>-</td>
<td class="tt_bl tt_b">+</td>
<td>+</td>
</tr>
<td>C</td>
<tr>
</tr>
<td class="tt_r">-</td>
<tr>
<td class="tt_b">+</td>
<td>0</td>
<td class="tt_bl tt_g">0</td>
<td>-</td>
</tr>
<td>C</td>
<tr>
</tr>
<td class="tt_g">0</td>
<tr>
<td class="tt_r">-</td>
<td>0</td>
<td class="tt_bl tt_b">+</td>
<td>0</td>
</tr>
<td>C</td>
<tr>
</tr>
<td class="tt_g">0</td>
<tr>
<td class="tt_g">0</td>
<td>0</td>
<td class="tt_bl tt_g">0</td>
<td>+</td>
</tr>
<td>C</td>
<tr>
</tr>
<td class="tt_g">0</td>
<tr>
<td class="tt_b">+</td>
<td>+</td>
<td class="tt_bl tt_r">-</td>
<td>-</td>
</tr>
<td>C</td>
<tr>
</tr>
<td class="tt_b">+</td>
<tr>
<td class="tt_r">-</td>
<td>+</td>
<td class="tt_bl tt_g">0</td>
<td>0</td>
</tr>
<td>C</td>
<tr>
</tr>
<td class="tt_b">+</td>
<tr>
<td class="tt_g">0</td>
<td>+</td>
<td class="tt_bl tt_r">-</td>
<td>+</td>
</tr>
<td>-</td>
<tr>
</tr>
<td class="tt_b">+</td>
</table>
<td class="tt_b">+</td>
<td class="tt_bl tt_b">+</td>
</tr>
</table>
</div>
</div>
[[Category:Ternary]]
[[Category:Logic_Gates]]

Latest revision as of 18:54, 21 January 2025

Modulo-3 Sum

Sum Gate Symbol
BCT Sum Gate

Uses

While there is a Ternary XOR gate, it's usefulness does not fit with what the Binary XOR can do. Another gate is needed to perform ternary addition.

SUM does mod 3 addition, returning the remainder.

It is also useful for cryptography and error correction being balanced, self-negating, associative, and commutative.

A ⊕ B = C
C ⊕ A = B
A ⊕ C = B
B ⊕ C = A
A ⊕ 0 = A
0 ⊕ A = A

Truth Tables

SUM

SUM B
- 0 +
A - + - 0
0 - 0 +
+ 0 + -
SUM
A B Y
- - +
- 0 -
- + 0
0 - -
0 0 0
0 + +
+ - 0
+ 0 +
+ + -

NSUM

NSUM B
- 0 +
A - - + 0
0 + 0 -
+ 0 - +
NSUM
A B Y
- - -
- 0 +
- + 0
0 - +
0 0 0
0 + -
+ - 0
+ 0 -
+ + +