vault backup: 2025-01-14 18:32:16

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education/computer engineering/ECE2700/Binary Logic.md

@@ -1,3 +1,56 @@
+# 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

education/computer engineering/ECE2700/Digital Hardware.md

@@ -47,7 +47,7 @@ As an example, the below table shows how one might convert from $(857)_{10}$ to
 | $3 / 2 = 1$     | $1$       |
 | $1 / 2 = 0$     | $1$       |
 
-The final answer is $1101011001$.
+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.
 # Definitions
 - **Xtor** is an abbreviation for *transistor*
 - **Moore's Law** states that the number of transistors on a chip doubles every two years.