In Boolean Algebra, the power to manipulate truth values comes from the fundamental operations: AND, OR, and NOT. These operations allow us to combine and modify boolean values to create more complex logic. Whether you’re designing a circuit, writing a program, or just solving logic puzzles, these basic operations will be your go-to tools. In this lesson, we’ll break down each operation, explore how they work with truth tables, and look at real-world examples where they are applied.

Introduction

Boolean operations are the heart of Boolean Algebra — they’re the building blocks used to create complex expressions, solve logical problems, and design digital systems. The three primary operations in Boolean Algebra are AND, OR, and NOT. Each operation has distinct rules and behaviors, allowing us to combine and manipulate truth values (True/False or 1/0) in meaningful ways.

• The AND operation checks if multiple conditions are true at the same time.
• The OR operation checks if at least one of multiple conditions is true.
• The NOT operation inverts a condition, changing true to false and vice versa.

These operations are crucial for creating logic circuits, programming conditional statements, building search filters, and many other real-world applications. Understanding how each of these works will lay the foundation for more advanced concepts, including logic gates, simplification of Boolean expressions, and digital circuit design.

The AND Operation

The AND operation is one of the most fundamental logical tools in Boolean Algebra. It takes two inputs and returns true only if both inputs are true. If either input is false — or both are false — the result is false.

In Boolean expressions, the AND operation is often represented in one of the following ways:

• A AND B
• A ∧ B
• A · B (dot notation)
• Or simply AB

All of these mean the same thing: both A and B must be true for the result to be true.

A	B	A AND B
0	0	0
0	1	0
1	0	0
1	1	1

This table shows all possible combinations of two boolean inputs, A and B, and the result of applying the AND operation.

Example

Let’s say we have two Boolean variables:

isRaining = 1 (true)

haveUmbrella = 1 (true)

We can write a logic expression:

stayDry = isRaining AND haveUmbrella

This means: you’ll stay dry only if it’s raining, and you have an umbrella. If you forget your umbrella (0), or it’s not raining (0), you won’t stay dry (0).

The OR Operation

The OR operation is another key element of boolean logic. It evaluates two inputs and returns true if at least one input is true. It only returns false if both inputs are false.

In boolean expressions, the OR operation is commonly written as:

• A OR B
• A ∨ B
• A + B (plus notation, not arithmetic addition)

No matter how it’s written, the logic is the same: if either A or B (or both) are true, the result is true.

A	B	A OR B
0	0	0
0	1	1
1	0	1
1	1	1

The truth table above shows that the only time OR gives a false result is when both inputs are false.

Example

Let’s say:

hasKeycard = 1 (true) knowsPassword = 0 (false)

We write the expression:

canAccess = hasKeycard OR knowsPassword

In this case, since you have a keycard, access is granted (1 OR 0 = 1). You don’t need both credentials — either one is enough.

This kind of logic is common in access control, software permissions, and filter settings (e.g., “show me red OR blue items”).

The NOT Operation

The NOT operation is different from AND and OR because it only works on one input. It simply inverts the value:

• If the input is true (1), the result is false (0).
• If the input is false (0), the result is true (1).

In boolean expressions, NOT is written in various ways:

• NOT A
• ¬A
• !A

As you can see in the truth table below, the NOT operation flips the truth value.

A	NOT A
0	1
1	0

Example

Let’s say:

isNight = 1 (true) isDay = NOT isNight

In this case, since it is night (1), then:

isDay = NOT 1 = 0

This kind of logic is useful when one condition is simply the opposite of another, like:

• Muted / Not Muted
• Enabled / Disabled
• Visible / Hidden

Key Takeaways

• AND (A AND B): Returns true only if both inputs are true.
• OR (A OR B): Returns true if at least one input is true.
• NOT (NOT A): Inverts the value of the input — true becomes false and vice versa.
• Each operation can be represented with truth tables, real-world analogies, and is used in programming, electronics, automation, and beyond.
• Mastering these operations is essential before moving on to more advanced concepts like expression simplification, standard forms, and circuit design.