> For the complete documentation index, see [llms.txt](https://cs.arkb.me/llms.txt). Markdown versions of documentation pages are available by appending `.md` to page URLs; this page is available as [Markdown](https://cs.arkb.me/hardware-software/boolean-algebra.md).

# Boolean Algebra

Just like normal algebra, Boolean algebra uses letters to represent values and uses them as expressions. Boolean algebra uses the values TRUE and FALSE (i.e. 1 and 0).

## Notation

<table><thead><tr><th width="374">Expression</th><th>Meaning</th></tr></thead><tbody><tr><td>A, B, C, etc.</td><td>Used to represent an unknown value, just like <em>x</em> or <em>y</em> in normal algebra.</td></tr><tr><td><span class="math">\overline A</span></td><td>NOT A. An overline represents the NOT operation being applied to what is below the line.</td></tr><tr><td><span class="math">A \cdot B</span></td><td>A AND B. The dot represents AND operation.</td></tr><tr><td><span class="math">A  + B</span></td><td>A OR B. The plus sign represents OR operation.</td></tr></tbody></table>

## Order of Precedence

The following list shows the order of precedence (Highest to Lowest):

* Brackets
* NOT
* AND
* OR

## Boolean Identities

There are numerous Boolean identities that can help *simplify* Boolean expressions:

$$
A \cdot 0 = 0
$$

$$
B \cdot 1 = B
$$

$$
C \cdot C = C
$$

$$
D + 0 = D
$$

$$
E + 1 = 1
$$

$$
F + F = F
$$

$$
\overline{\overline{C}} = C
$$

## De Morgan's Law

De Morgan's law can be very useful when simplifying logical expressions. It can be remembered by recalling the phrase:

> Break the bar and change the sign

The "bar" refers to the overline representing the NOT sign and the sign refers to changing the dot or plus sign.&#x20;

Example of using De Morgan's law on $$\overline{A + B}$$:&#x20;

* Step 1: Break the bar $$\overline{A} + \overline{B}$$
* Step 2: Change the sign: $$\overline{A} \cdot \overline{B}$$

De Morgan's law can also be used in reverse, by changing the sign and building the bar. Following is an example of changing $$\overline{C} + \overline{D}$$:

* Step 1: Change the sign $$\overline{C} \cdot \overline{D}$$
* Step 2: Build the bar $$\overline{C \cdot D}$$

## Distributive Rules

Just like mathematics, you can expand and factorise Boolean expressions.

$$
A \cdot (B + C) = A \cdot B + A \cdot C
$$

## Example Questions

<details>

<summary>Simplify the Boolean expression <span class="math">A + \overline{B\cdot A}</span></summary>

1. Use De Morgan's Law to break bar $$A + \overline{B} + \overline{A}$$
2. Use $$A + \overline{A} = 1$$ -> $$\overline{B} + 1$$
3. \= 1

</details>

<details>

<summary>Simplify the Boolean expression <span class="math">C \cdot B + \overline{C} \cdot B</span></summary>

1. Factorise: $$B \cdot (C + \overline{C})$$&#x20;
2. $$= B \cdot (1) = B$$

</details>


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