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 energy, and less delay when the gate path gets shorter.
What a learner can do afterwards
- 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
1 · Read
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).
De Morgan's law turns NOT over an AND into the OR of the flipped sides: not (A and B) equals (not A) or (not B). To check a rewrite, fill in a truth table for both forms across all four rows of A and B. Every row must match. If even one row differs, the rewrite is wrong.
Simplifying swaps a long expression for a shorter one with the same truth table. When two AND terms share an input, it can come outside: (A and B) or (A and C) equals A and (B or C). Counting each two-input AND or OR as one gate, the long form uses 3 gates and the short form uses 2, so 1 gate is saved. Fewer gates means less silicon and less energy. Delay drops only when the longest path through the gates gets shorter.
A written rule becomes gates in three moves: write it as an expression, shrink it, then give each operation a gate. An alarm sounds only when the door is open and the system is armed. It stays quiet when not (open and armed), which De Morgan turns into (not open) or (not armed). Check every row of the truth table before you build.
NOT before AND before OR. De Morgan flips the sides and swaps the operator. Shrink, check every row, then build with fewer gates.
Lesson text, worked examples, flashcards and practice questions with worked steps, written for this dataset.
Four readings written for this dataset. Two videos are linked from the academy collection: Circuits & Logic, and Evaluating compound boolean expressions.
2 · Watch
En anglaisTake it off screen
Where it sits
Learn first
Optional connections
This opens up
8 questions wait behind this lesson, each with its answer explained. Every answer feeds the sky: stars light as they are learned, and dim when it is time to come back.