---
# System prepended metadata

title: Complete Boolean Truth Tables
tags: [boolean, math, logic]

---

---
title: "Complete Boolean Truth Tables"
path: "Complete Boolean Truth Tables"
---

{%hackmd @RintarouTW/About %}

# Complete Boolean Truth Tables

Binary Operator Truth Tables

|     | Expression                                                                                                                                                                      | Inverse                                                                                                                                                                         |
| --- | ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- | ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- |
| 1   | $1$                                                                                                                                                                             | $0$                                                                                                                                                                             |
| 2   | $A$                                                                                                                                                                             | $!A$                                                                                                                                                                            |
| 3   | $B$                                                                                                                                                                             | $!B$                                                                                                                                                                            |
| 4   | $\left\{\begin{array}{c}A \cdot B，皆通方通/皆是方是\\!(!A + !B)，有斷則斷/有否則否\end{array}\right.$                                                                             | $\left\{\begin{array}{c}!(A \cdot B)，皆通方斷/皆是方否\\!A + !B，有斷則通/有否則是\end{array}\right.$                                                                             |
| 5   | $\left\{\begin{array}{cc}A + B，皆斷方斷/皆否方否\\!(!A \cdot !B)，有通則通/有是則是\end{array}\right.$                                                                            | $\left\{\begin{array}{cc}!(A + B)，皆斷方通/皆否方是\\!A \cdot !B，有通則斷/有是則否\end{array}\right.$                                                                            |
| 6   | $\left\{\begin{array}{c}!A \cdot B，01 方通\\!(A + !B)，非 01 則斷\end{array}\right.$                                                                                              | $\left\{\begin{array}{c}!(!A \cdot B)，01 方斷\\A + !B，非 01 則通\end{array}\right.$                                                                                              |
| 7   | $\left\{\begin{array}{c}A \cdot !B，10 方通\\!(!A + B)，非 10 則斷\end{array}\right.$                                                                                              | $\left\{\begin{array}{c}!(A \cdot !B)，10 方斷\\!A + B，非 10 則通\end{array}\right.$                                                                                              |
| 8   | $\left\{\begin{array}{c}(A \cdot !B) + (!A \cdot B)\\相異則通/相異則是\\![(A + !B) \cdot (!A + B)]\\相同則斷/相同則否\\![(A \cdot B) + (!A \cdot !B)]\\(A + B) \cdot (!A + !B)\end{array}\right.$ | $\left\{\begin{array}{c}![(A \cdot !B) + (!A \cdot B)]\\相異則斷/相異則否\\(A + !B) \cdot (!A + B)\\相同則通/相同則是\\(A \cdot B) + (!A \cdot !B)\\![(A + B) \cdot (!A + !B)]\end{array}\right.$ |

1) 1 , 0

| $A$ | $B$ | $C$ | $\bar{C}$ |
| --- | --- | --- | --------- |
| 0   | 0   | 1   | 0         |
| 0   | 1   | 1   | 0         |
| 1   | 0   | 1   | 0         |
| 1   | 1   | 1   | 0         |

<img src="https://i.imgur.com/YLOlfHn.png" style="filter:invert(.9)"/><br>

<img src="https://i.imgur.com/H2PJAEx.png" style="filter:invert(.9)"/>


2) OR (皆斷方斷/有通則通)，NOR (皆斷方通/有通則斷)

| $A$ | $B$ | $C$ | $\bar{C}$ |
| --- | --- | --- | --------- |
| 0   | 0   | 0   | 1         |
| 0   | 1   | 1   | 0         |
| 1   | 0   | 1   | 0         |
| 1   | 1   | 1   | 0         |

<img src="https://i.imgur.com/VXZl5Lf.png" style="filter:invert(.9)"/><br>

<img src="https://i.imgur.com/qIeAbsC.png" style="filter:invert(.9)"/>


3) (A + !B), (!A · B)

| $A$ | $B$ | $C$ | $\bar{C}$ |
| --- | --- | --- | --------- |
| 0   | 0   | 1   | 0         |
| 0   | 1   | 0   | 1         |
| 1   | 0   | 1   | 0         |
| 1   | 1   | 1   | 0         |

<img src="https://i.imgur.com/acp41Jb.png" style="filter:invert(.9)"/><br>

<img src="https://i.imgur.com/4d2ouBG.png" style="filter:invert(.9)"/>


4) A, !A

| $A$ | $B$ | $C$ | $\bar{C}$ |
| --- | --- | --- | --------- |
| 0   | 0   | 0   | 1         |
| 0   | 1   | 0   | 1         |
| 1   | 0   | 1   | 0         |
| 1   | 1   | 1   | 0         |

<img src="https://i.imgur.com/gQRNgHq.png" style="filter:invert(.9)"/><br>

<img src="https://i.imgur.com/M6phkxA.png" style="filter:invert(.9)"/>

5) (!A + B), (A · !B)

| $A$ | $B$ | $C$ | $\bar{C}$ |
| --- | --- | --- | --------- |
| 0   | 0   | 1   | 0         |
| 0   | 1   | 1   | 0         |
| 1   | 0   | 0   | 1         |
| 1   | 1   | 1   | 0         |

<img src="https://i.imgur.com/xAVDd5i.png" style="filter:invert(.9)"></img><br>

<img src="https://i.imgur.com/KU2MpiJ.png" style="filter:invert(.9)"></img>

6) B , !B

| $A$ | $B$ | $C$ | $\bar{C}$ |
| --- | --- | --- | --------- |
| 0   | 0   | 0   | 1         |
| 0   | 1   | 1   | 0         |
| 1   | 0   | 0   | 1         |
| 1   | 1   | 1   | 0         |

<img src="https://i.imgur.com/TlP2Pp6.png" style="filter:invert(.9)"></img><br>

<img src="https://i.imgur.com/PjUTftw.png" style="filter:invert(.9)"></img>

7) XNOR (相異則斷/相同則通), XOR (相異則通/相同則斷)

| $A$ | $B$ | $C$ | $\bar{C}$ |
| --- | --- | --- | --------- |
| 0   | 0   | 1   | 0         |
| 0   | 1   | 0   | 1         |
| 1   | 0   | 0   | 1         |
| 1   | 1   | 1   | 0         |

<img src="https://i.imgur.com/CcQf1iG.png" style="filter:invert(.9)"></img><br>

<img src="https://i.imgur.com/MhYd5GT.png" style="filter:invert(.9)"></img>

8) AND (皆通方通/有斷則斷), NAND (皆通方斷/有斷則通)

| $A$ | $B$ | $C$ | $\bar{C}$ |
| --- | --- | --- | --------- |
| 0   | 0   | 0   | 1         |
| 0   | 1   | 0   | 1         |
| 1   | 0   | 0   | 1         |
| 1   | 1   | 1   | 0         |

<img src="https://i.imgur.com/C6HwiEY.png" style="filter:invert(.9)"></img><br>

<img src="https://i.imgur.com/yaLPPRI.png" style="filter:invert(.9)"></img>

## Logic Operations

0 + X = X
0 · X = 0
1 + X = 1
1 · X = X
X + !X = 1
X · !X = 0

A · B = !(!A + !B)
A + B = !(!A · !B)
!A + B = !(A · !B)
!A · B = !(A + !B)
A + !B = !(!A · B)
A · !B = !(!A + B)

A XOR B = (!A · B) + (A · !B)
        = (A + B) · (!A + !B)

A XNOR B = (A + !B) · (!A + B)
         = (A · B) + (!A · !B)

## Set

<img src="https://i.imgur.com/ShGgpBb.png" style="filter:invert(.9)"></img>


###### tags: `boolean` `math` `logic`
