Discrete Mathematics · Quick Reference

Boolean Algebra Cheat Sheet

Discrete Mathematics · Lesson 7/9

In one line: boolean algebra deals with variables that take only two values — true/false or 1/0 — forming the mathematical basis for digital logic circuits and computer processing.

Key Ideas

1Boolean Variables. Variables that can only take the value 0 (false) or 1 (true).

2Basic Operations. AND (multiplication-like, written as a dot or juxtaposition), OR (addition-like, written as +), NOT (complement, written with an overbar or apostrophe).

3Boolean Identities. Key rules like A + 0 = A, A * 1 = A, A + A' = 1, A * A' = 0, where A' means NOT A.

4De Morgan's Laws. NOT(A AND B) = (NOT A) OR (NOT B), and NOT(A OR B) = (NOT A) AND (NOT B) — flipping AND/OR while negating each term.

5Logic Gates. Physical or simulated circuits implementing AND, OR, NOT, and combinations like NAND, NOR, and XOR — the building blocks of digital electronics.

Worked Examples

Simplify A + A*B using Boolean algebra (absorption law).

A + A*B = A

Apply De Morgan's Law to simplify NOT(A AND B) where A=1, B=0.

Result = 1 (confirmed by both direct evaluation and De Morgan's Law)

Evaluate the Boolean expression A*B + A'*C when A=1, B=0, C=1.

Result = 0

Formulas

A + A' = 1

A * A' = 0

De Morgan: NOT(A AND B) = (NOT A) OR (NOT B)

De Morgan: NOT(A OR B) = (NOT A) AND (NOT B)

Practice Yourself

Evaluate A + A' for any value of A.

1 (always true)

Evaluate A * A' for any value of A.

0 (always false)

Simplify A * 1.