Logic in Circuit Design

Mathematical Logic · Axiom Academy

REAL WORLD Logic in Circuit Design Discover how propositional logic powers every computer, smartphone, and digital device through logic gates and Boolean algebra Every processor in your computer, smartphone, and smart watch is built from billions of tiny switches called transistors . These transistors implement logic gates —physical realizations of the logical operators you've been studying in Mathematical Logic! Physical electronic components In digital circuits, we use voltage levels to represent truth values. Typically, HIGH voltage (like 5V or 3.3V) represents TRUE (1), and LOW voltage (0V) represents FALSE (0). Logic gates are electronic circuits that perform logical operations on these voltage signals. Let's explore the basic building blocks of all digital circuits. Each gate implements a specific logical operation. Interactive Logic Gate Simulator Now let's see these gates in action! Toggle the inputs and watch how the output changes based on the logical operation. Try this: Set both inputs to 1 on the NAND gate. What do you notice? Boolean Algebra and Circuit Simplification Just as you learned logical equivalences in propositional logic, Boolean algebra provides rules for simplifying circuits. These equivalences allow engineers to reduce the number of gates needed, saving space, power, and cost! De Morgan's: ¬(P ∧ Q) ≡ ¬P ∨ ¬Q Distributive: P ∧ (Q ∨ R) ≡ (P ∧ Q) ∨ (P ∧ R) De Morgan's: (A · B)' = A' + B' Distributive: A · (B + C) = A·B + A·C

This is the written version of the interactive lesson above. See the full Mathematical Logic course.