---
title: "Boolean Algebra and Simplifying Logic Expressions"
description: "Writing a circuit as an algebraic expression and rearranging it with the laws of Boolean algebra, including De Morgan's, to reach a shorter expression with the same truth table. Fewer gates means less"
canonical: https://lightmysky.com/learn/computing/boolean-algebra-and-simplifying-logic-expressions-mt_9MrE1N63D5
source: https://lightmysky.com/learn/computing/boolean-algebra-and-simplifying-logic-expressions-mt_9MrE1N63D5.md
retrieved: 2026-09-12
---

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# Boolean Algebra and Simplifying Logic Expressions

Writing a circuit as an algebraic expression and rearranging it with the laws of Boolean algebra, including De Morgan's, to reach a shorter expression with the same truth table. Fewer gates means less silicon and less delay.

Subject: Computing · Area: Computer Systems · Ages 16 to 17
Page: https://lightmysky.com/learn/computing/boolean-algebra-and-simplifying-logic-expressions-mt_9MrE1N63D5

## Ready when they can

- Rewrite NOT over an AND using De Morgan's laws and check the result against a truth table
- Simplify an expression of three variables and count the gates saved
- Turn a written condition into an expression and then into gates

## Lesson: Shorter logic, same truth

Three operations build every logic expression. AND needs both sides true, OR needs at least one side true, and NOT flips its input. When operators mix, NOT runs first, then AND, then OR, while brackets override everything. So not A or B means (not A) or B, never not (A or B).

**Example.** De Morgan law turns NOT over an AND into the OR of the flipped sides: not (A and B) equals (not A) or (not B). Check it the honest way by building a truth table for both forms across all four rows. Every row must match, and if a single row differs, the rewrite is wrong.

Simplification swaps a long expression for a short one with the same truth table. If the long form needs three gates and the short form needs one, you saved two gates. Fewer gates means less silicon, less delay and less energy, while the circuit answers exactly as before.

**Tip.** Turn a written condition into gates in three moves. First write the condition as an expression, then shrink it with laws like De Morgan, then map each remaining operation to a gate. Never skip the truth table check between shrinking and building.

**Recap.** Flip the sides and swap the operator, prove it on every row, then build with fewer gates.

## Practice

8 questions on this page, each with its working shown.

## Needs first

- [Boolean Expressions and Comparison Operators](https://lightmysky.com/learn/computing/boolean-expressions-and-comparison-operators-mt_0aIh-DPCZ4)
- [Logic Gates and Truth Tables](https://lightmysky.com/learn/computing/logic-gates-and-truth-tables-mt_o2N0-q3A4Q)

## Opens up

- [Logic Circuits That Add and Remember](https://lightmysky.com/learn/computing/logic-circuits-that-add-and-remember-mt_zPgkt6zf-i)