Draw logic circuits based on problem statements, logic expressions or truth tables

Resources | Subject Notes | Computer Science

Boolean Logic

Introduction

Boolean logic is the foundation of digital circuit design and computer science. It deals with logical operations that can produce true (1) or false (0) outputs. The core concepts involve Boolean algebra, logic gates, and their applications in building complex circuits.

Basic Boolean Operators

There are three primary Boolean operators:

  • AND (Suggested diagram: AND gate): The output is 1 only if both inputs are 1. Symbol: $\land$ or $\cdot$
  • OR (Suggested diagram: OR gate): The output is 1 if at least one input is 1. Symbol: $\lor$ or $+$
  • NOT (Suggested diagram: NOT gate): The output is the inverse of the input. If the input is 1, the output is 0, and vice versa. Symbol: $eg$ or $\overline{x}$

Truth Tables

A truth table shows all possible input combinations and their corresponding output for a Boolean expression.

Here's a truth table for the AND operation:

Input A Input B Output (A $\land$ B)
0 0 0
0 1 0
1 0 0
1 1 1

Here's a truth table for the OR operation:

Input A Input B Output (A $\lor$ B)
0 0 0
0 1 1
1 0 1
1 1 1

Here's a truth table for the NOT operation:

Input Output ($eg$ A)
0 1
1 0

Logic Gates

Logic gates are electronic circuits that implement Boolean logic operations.

  1. AND Gate: Outputs 1 only when both inputs are 1.
  2. OR Gate: Outputs 1 if at least one input is 1.
  3. NOT Gate: Inverts the input.
  4. NAND Gate: The output is the inverse of the AND gate.
  5. NOR Gate: The output is the inverse of the OR gate.
  6. XOR Gate: Outputs 1 if the inputs are different.

Drawing Logic Circuits

To draw a logic circuit:

  1. Identify the Boolean expression.
  2. Determine the required logic gates based on the expression.
  3. Draw the gates and connect them according to the expression.
  4. Label the inputs and outputs of the gates.

Examples

Example 1: Expressing a Boolean Function

Consider the Boolean function: $F = (A \land B) \lor (eg C)$

This function can be implemented using logic gates as follows:

Suggested diagram: A circuit with an AND gate (inputs A and B), an OR gate (inputs from the AND gate and the NOT gate output of C).

The circuit would consist of a NOT gate for C, an AND gate for A and B, and an OR gate for the combined outputs.

Example 2: Drawing a Logic Circuit from a Problem Statement

Problem: Design a circuit that outputs 1 if and only if both inputs are 0.

Solution: This is equivalent to the AND operation. Therefore, we need an AND gate with inputs for the two variables.

Suggested diagram: A circuit with an AND gate (inputs for two variables).

Further Study

For a deeper understanding, explore concepts like Boolean algebra simplification, Karnaugh maps, and more complex logic circuits.