notes/education/computer engineering/ECE2700/Binary Logic.md

# Properties of Boolean Algebra
| Number                 | Col. A                                      | Col. A Description | Col. B                                 | Col. B Description |
| ---------------------- | ------------------------------------------- | ------------------ | -------------------------------------- | ------------------ |
| 1.                     | $0 \cdot 0 = 0$                             |                    | $1 + 1 = 1$                            |                    |
| 2.                     | $1 \cdot 1 = 1$                             |                    | $0 + 0 = 0$                            |                    |
| 3.                     | $0 \cdot 1 = 1 \cdot 0 = 0$                 |                    | $1 + 0 = 0 + 1 = 1$                    |                    |
| 4.                     | if $x = 0$ then $\overline{x} = 1$          |                    | if $x = 1$ then $\overline{x} = 0$     |                    |
| 5.                     | $x \cdot 0 = 0$                             |                    | $x + 1 = 1$                            |                    |
| 6.                     | $x \cdot 1 = x$                             |                    | $x + 0 = x$                            |                    |
| 7.                     | $x \cdot x = x$                             |                    | $x + x = x$                            |                    |
| 8.                     | $x \cdot \overline{x} = 0$                  |                    | $$x + \overline{x} = 1$                |                    |
| 9.                     | $\overline{\overline{x}} = x$               |                    |                                        |                    |
| 10. Commutative        | $x \cdot y = y \cdot x$                     |                    | $x + y = y + x$                        |                    |
| 11. Associative        | $x \cdot (y \cdot z) = (x \cdot y) \cdot z$ |                    | $x + (y + z) = (x + y) +z$             |                    |
| 12. Distributive       | $x \cdot (y +z) = x \cdot y + x \cdot z$    |                    | $x + y \cdot z = (x + y) \cdot (x + z$ |                    |
| 13. Absorption         | $x + x \cdot y                              |                    |                                        |                    |
| 14. Combining          |                                             |                    |                                        |                    |
| 15. DeMorgan's Theorem |                                             |                    |                                        |                    |
| 16.                    |                                             |                    |                                        |                    |
| 17. Consensus          |                                             |                    |                                        |                    |

# Logic Gates

![](./assets/logic-gates.jpeg)
# NOT Gate
A binary NOT gate has a single input, and inverts that input (output is not the input).

## Truth Table
| $x$ | $y$ |
| --- | --- |
| 0   | 1   |
| 1   | 0   |
## Mathematical Expression
A NOT operation is mathematically expressed using a bar:
$$ y = \bar{x} $$
# AND Gate
An AND gate will only output a 1 if *both* inputs are a one (input one *and* input two are enabled).

## Truth Table
| $x_1$ | $x_2$ | $y$ |
| ----- | ----- | --- |
| 0     | 0     | 0   |
| 0     | 1     | 0   |
| 1     | 0     | 0   |
| 1     | 1     | 1   |
## Mathematical Expression
An AND operation is mathematically expressed using a times symbol, or with no symbol at all:
$$ y = x_1 \cdot x_2 = x_1x_2$$

# NAND Gate
A NAND gate outputs a 1 *unless* both inputs are enabled (input one *and* input two are *not* enabled).

## Truth Table
| $x_1$ | $x_2$ | $y$ |
| ----- | ----- | --- |
| 0     | 0     | 1   |
| 0     | 1     | 1   |
| 1     | 0     | 1   |
| 1     | 1     | 0   |
## Mathematical Expression
A NAND operation is mathematically expressed using a bar over an AND operation:
$$ y = \overline{x_1 \cdot x_2}$$


# OR Gate
An OR gate outputs a 1 if either or both inputs are enabled (if input one *or* input two is enabled).
## Truth Table
| $x_1$ | $x_2$ | $y$ |
| ----- | ----- | --- |
| 0     | 0     | 0   |
| 0     | 1     | 1   |
| 1     | 0     | 1   |
| 1     | 1     | 1   |
## Mathematical Expression
A mathematical OR is notated with a $+$ symbol.

$$ y = x_1 + x_2 $$
# NOR Gate
A NOR gate outputs a one if neither gate is enabled.
## Truth Table
| $x_1$ | $x_2$ | $y_1$ |
| ----- | ----- | ----- |
| 0     | 0     | 1     |
| 0     | 1     | 0     |
| 1     | 0     | 0     |
| 1     | 1     | 0     |
## Mathematical Expression
A NOR operation is expressed using a bar over an OR operation.
$$ y = \overline{x_1 + x_2} $$
# XOR Gate
An XOR gate is on if one input is enabled, but *not* both (exclusively one or the other).

## Truth Table
| $x_1$ | $x_2$ | $y$ |
| ----- | ----- | --- |
| 0     | 0     | 0   |
| 0     | 1     | 1   |
| 1     | 0     | 1   |
| 1     | 1     | 0   |
## Mathematical Expression
An XOR operation is expressed using a circle around an addition symbol:
$$ y = x_1 \oplus x_2 $$

## XNOR Gate
An XNOR gate is on if neither input is enabled, or both inputs are enabled.

## Truth Table

| $x_1$ | $x_2$ | $y$ |
| ----- | ----- | --- |
| 0     | 0     | 1   |
| 0     | 1     | 0   |
| 1     | 0     | 0   |
| 1     | 1     | 1   |
## Mathematical Expression
An XNOR operation is expressed using a bar over an XOR operation:
$$ y = \overline{x_1 \oplus x_2} $$