Question

Give the Boolean expression for a three-input AND gate, and then show its behavior with a truth table.

Short answer

The Boolean expression is \( Y = A \cdot B \cdot C \), and it outputs 1 only when all inputs are 1.

Step-by-step solution

Understand the Three-Input AND Gate

A three-input AND gate will output TRUE (or 1) only if all three inputs are TRUE (or 1). The inputs can be represented as \( A \), \( B \), and \( C \).

Write the Boolean Expression

For an AND gate with inputs \( A \), \( B \), and \( C \), the Boolean expression is:\[ Y = A \cdot B \cdot C \]Here, \( Y \) is the output, and \( \cdot \) represents the AND operation.

Create the Truth Table

A truth table lists all possible input combinations and the corresponding outputs for the Boolean expression:| \( A \) | \( B \) | \( C \) | \( Y = A \cdot B \cdot C \) ||---|---|---|---|| 0 | 0 | 0 | 0 || 0 | 0 | 1 | 0 || 0 | 1 | 0 | 0 || 0 | 1 | 1 | 0 || 1 | 0 | 0 | 0 || 1 | 0 | 1 | 0 || 1 | 1 | 0 | 0 || 1 | 1 | 1 | 1 |In this table, each row represents a different combination of \( A \), \( B \), and \( C \), and the value in the last column is the output \( Y \) for each combination. As shown, the output is 1 only if all inputs are 1.

Key concepts

AND Gate

An AND gate is a fundamental building block in digital electronics. Imagine it like a team approval process. Everyone on the team needs to say "yes" for the task to proceed. Similarly, in an AND gate, all inputs must be true for the output to be true. In simpler terms, it performs a multiplication-like operation on the inputs.
An AND gate checks multiple conditions at once, and it's only satisfied when all those conditions are met. Whether you're building circuits or solving logically complex problems, AND gates are indispensable for handling multiple inputs effectively. They ensure all necessary criteria are satisfied before allowing the circuit to proceed or activate.

Truth Table

Truth tables are an essential tool for understanding how logical operations like AND work. They are like decision-making charts that outline all possible scenarios for inputs and their corresponding outputs.
To build a truth table for a three-input AND gate, you must list every combination of inputs and compute the output for each. Here's how it goes:

• Each input can be 0 (false) or 1 (true).
• With three inputs - let's call them \( A \), \( B \), and \( C \) - there are \( 2^3 = 8 \) possible combinations.

For AND gates, outputs follow a strict rule. The output is true (1) only if every input among \( A \), \( B \), and \( C \) is true together. Think of a lock that only opens when all the pins line up perfectly. If even one is out of place (i.e., is 0), the circuit remains closed, producing a false (0) output.

Boolean Expression

In digital logic, Boolean expressions are the math-like representations of logic operations. For an AND gate with three inputs, the Boolean expression defines exactly how it functions: \( Y = A \cdot B \cdot C \).
Each letter in the equation denotes an input variable, while the \( \cdot \) symbol represents the AND operation, akin to multiplication in arithmetic. When interpreting this equation:

• \( Y \) is the resulting output.
• The presence of the dot shows that inputs are all jointly required for a positive result.

By understanding Boolean expressions, you can not only design complex logical circuits with multiple layers but also simplify and troubleshoot existing systems. Essentially, like a concise rulebook, they guide how inputs should interact to produce desired outputs, which is crucial in both software and hardware design.