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education/computer engineering/ECE2700/Binary Logic.md
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# History of Boolean Algebra
- In 1849, George Boole published a scheme for describing logical thought and reasoning
- In the 1930s, Claude Shannon applied Boolean algebra to describe circuits built with switches
- Boolean algebra provides the theoretical foundation for digital design
# 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. | $x \cdot \overline{x} = 0$ | | $$x + \overline{x} = 1$ | |
| 9. | $\overline{\overline{x}} = x$ | | | |
| 10. Commutative | $x \cdot y = y \cdot x$ | | $x + y = y + x$ | |
| 11. Associative | $x \cdot (y \cdot z) = (x \cdot y) \cdot z$ | | $x + (y + z) = (x + y) +z$ | |
| 12. Distributive | $x \cdot (y +z) = x \cdot y + x \cdot z$ | | $x + y \cdot z = (x + y) \cdot (x + z$ | |
| 13. Absorption | $x + x \cdot y = x$ | | $x \cdot (x + y) = x$ | |
| 14. Combining | $x \cdot y + x \cdot \overline{y} = x$ | | $(x + y) \cdot (x + \overline{y}) = x$ | |
| 15. DeMorgan's Theorem | $\overline{x \cdot y} = \overline{x} + \overline{y}$ | | $x + y = \overline{x} \cdot \overline{y}$ | |
| 16. | $x + \overline{x} \cdot y = x + y$ | | $x \cdot (\overline{x} + y) = x \cdot y$ | |
| 17. Consensus | $x \cdot y + y \cdot z + \overline{x} \cdot z = x \cdot y + \overline{x} \cdot z$ | | $(x + y) \cdot (y + z) \cdot (\overline{x} + z) = (x + y) \cdot (\overline{x} + z)$ | |
# Synthesis
In the context of binary logic, synthesis refers to the act of creating a boolean expression that evaluates to match a given truth table.
This is done by creating a product term for each entry in the table that has an output of $1$, that also evaluates to $1$, then ORing each product term together and then simplifying.
Example:
Given the below truth table, synthesize a boolean expression that corresponds.
| $x_1$ | $x_2$ | $f(x_1, x_2)$ |
| ----- | ----- | ------------- |
| 0 | 0 | 1 |
| 0 | 1 | 1 |
| 1 | 0 | 0 |
| 1 | 1 | 1 |
- $f(0, 0)$ evaluates to true with the expression $\overline{x}_1 \cdot \overline{x}_2$
- $f(0, 1)$ evaluates to true with the expression $\overline{x}_1\cdot x_2$
- $f(1, 0)$ should provide an output of zero, so that can be ignored
- $f(1, 1)$ evaluates to true with the expression $x_1 \cdot x_2$
ORing all of the above expression together, we get:
$$ f(x_1, x_2) = \overline{x}_1\overline{x}_2 + \overline{x}_1 x_2 + x_1x_2 $$
$$
\begin{multline}
= x_1x_2 \\
= x
\end{multline}
$$
# Logic Gates
![](./assets/logic-gates.jpeg)
# NOT Gate

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education/computer engineering/ECE2700/Digital Hardware.md
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| $3 / 2 = 1$ | $1$ |
| $1 / 2 = 0$ | $1$ |
The final answer is $1101011001$. The least significant bit is the remainder of the first division operation, and the most significant bit is the remainder of the last operation.
The final answer is $1101011001$.
# Definitions
- **Xtor** is an abbreviation for *transistor*
- **Moore's Law** states that the number of transistors on a chip doubles every two years.