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# Boolean algebra - Core rules (Simple & Practical)

Understanding Boolean algebra is essential for writing clean, reliable, and optimized code. These core logical laws form the foundation of condition design, query construction, filtering logic, and software architecture decisions.

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Ivan Borshchov

CEO & System Architect @ Devforth

Feb 12, 2026

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# [](#boolean-algebra-----core-rules-simple--practical)Boolean Algebra --- Core Rules (Simple & Practical)

Boolean algebra is a system for working with logical values:

-   `1` / `true`
-   `0` / `false`

It is used everywhere in programming: `if` statements, filtering, bit operations, SQL conditions, hardware logic, etc.

* * *

## [](#basic-operators)Basic Operators

| Operator | Symbol | Meaning | Scientific Naming (Discrete Math / Logic) |
| --- | --- | --- | --- |
| AND | a && b or a ∧ b | true if **both** are true | **Conjunction** |
| OR | a || b or a ∨ b | true if **at least one** is true | **Disjunction** |
| NOT | !a or ¬a | logical negation | **Negation** |

* * *

# [](#core-laws)Core Laws

## [](#1️⃣-idempotent-law)1️⃣ Idempotent Law

Repeating the same value changes nothing.

```
a && a = a
a || a = a
```

* * *

## [](#2️⃣-commutative-law)2️⃣ Commutative Law

Order does not matter.

```
a && b = b && a
a || b = b || a
```

* * *

## [](#3️⃣-associative-law)3️⃣ Associative Law

Grouping does not matter.

```
(a && b) && c = a && (b && c)
(a || b) || c = a || (b || c)
```

* * *

## [](#4️⃣-distributive-law-very-important)4️⃣ Distributive Law (Very Important)

```
a && (b || c) = (a && b) || (a && c)
a || (b && c) = (a || b) && (a || c)
```

* * *

## [](#5️⃣-identity-neutral-elements)5️⃣ Identity (Neutral Elements)

```
a && true  = a
a || false = a
```

* * *

## [](#6️⃣-domination-absorbing-elements)6️⃣ Domination (Absorbing Elements)

```
a && false = false
a || true  = true
```

* * *

## [](#7️⃣-complement-law)7️⃣ Complement Law

```
a && !a = false
a || !a = true
```

* * *

## [](#8️⃣-double-negation)8️⃣ Double Negation

```
!!a = a
```

* * *

## [](#9️⃣-de-morgans-laws-extremely-important)9️⃣ De Morgan's Laws (Extremely Important)

```
!(a && b) = !a || !b
!(a || b) = !a && !b
```

* * *

## [](#-absorption-law)🔟 Absorption Law

```
a || (a && b) = a
a && (a || b) = a
```

* * *

# [](#example-simplification)Example Simplification

```
!(a && b)
```

Using De Morgan:

```
!a || !b
```

* * *

# [](#what-is-used-most-in-programming)What Is Used Most in Programming

In real-world code, these rules are used most often:

-   **De Morgan's Laws**
-   **Double negation (`!!a`)**
-   **Complement law (`a && !a`)**
-   **Distributive law**
-   **Absorption law**

These are especially common when simplifying:

-   complex `if` conditions\\
-   SQL `WHERE` clauses\\
-   filtering logic\\
-   guard conditions\\
-   bitwise operations

* * *

Boolean algebra is not just theory --- it is the foundation of clean condition design.

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Table of Contents

-   [Basic Operators](#basic-operators)
-   [1️⃣ Idempotent Law](#_1️⃣-idempotent-law)
-   [2️⃣ Commutative Law](#_2️⃣-commutative-law)
-   [3️⃣ Associative Law](#_3️⃣-associative-law)
-   [4️⃣ Distributive Law (Very Important)](#_4️⃣-distributive-law-very-important)
-   [5️⃣ Identity (Neutral Elements)](#_5️⃣-identity-neutral-elements)
-   [6️⃣ Domination (Absorbing Elements)](#_6️⃣-domination-absorbing-elements)
-   [7️⃣ Complement Law](#_7️⃣-complement-law)
-   [8️⃣ Double Negation](#_8️⃣-double-negation)
-   [9️⃣ De Morgan's Laws (Extremely Important)](#_9️⃣-de-morgans-laws-extremely-important)
-   [🔟 Absorption Law](#absorption-law)

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