What do the symbols mean in Boolean algebra?

+ means OR, a dot or two letters written together means AND and an apostrophe, exclamation mark or overbar means NOT.

What is De Morgan’s law?

NOT (A AND B) equals NOT A OR NOT B and NOT (A OR B) equals NOT A AND NOT B. It converts between AND and OR when a NOT is applied across a bracket.

How do I prove two Boolean expressions are equal?

Either simplify both to the same form using the laws or build both truth tables and show every row matches.

What is sum of products form?

An expression written as ANDed terms joined by ORs, such as AB + A’C. It maps directly onto a two level logic circuit and is the standard target form for simplification.

Can this help with programming conditions?

Yes. Complex if statements are Boolean expressions, so simplifying them produces cleaner code. A common example: NOT (a AND b) rewrites as NOT a OR NOT b.

Specialist Calculator

Boolean Algebra Calculator

Name: Boolean Algebra Calculator
Author: CalZone

Simplifying a Boolean expression by hand means juggling half a dozen laws and hoping nothing slipped. Our boolean algebra calculator simplifies any logic expression step by step, generates full truth table and shows exactly which law was applied at each stage, so the working teaches you method rather than just handing over an answer.

Boolean Logic Solver

Input A

Input B

AND (A ∧ B)0

OR (A ∨ B)1

XOR (A ⊕ B)1

NOT A (¬A)0

NAND1

NOR0

NOT B1

Truth Table Hint

AND is only true if both are true. OR is true if at least one is true. XOR is true if exactly one is true (they must be different). NAND and NOR are the inverted versions of AND and OR.

How the Boolean Algebra Calculator Works

Type your expression using standard notation and choose whether you want it simplified, evaluated or turned into a truth table. The calculator parses variables, applies the laws of Boolean algebra in sequence and displays every transformation until the expression reaches its simplest equivalent form.

Common input notation: + for OR, . or simple adjacency for AND and an apostrophe or for NOT, so A’B + AB’ reads as NOT A AND B, OR A AND NOT B. Brackets group terms exactly as in ordinary algebra.

The Laws of Boolean Algebra

A small set of laws powers every simplification:

Identity: A + 0 = A and A . 1 = A
Null: A + 1 = 1 and A . 0 = 0
Idempotent: A + A = A and A . A = A
Complement: A + A’ = 1 and A . A’ = 0
Double negation: (A’)’ = A
Absorption: A + AB = A
Distributive: A(B + C) = AB + AC
De Morgan’s theorems: (AB)’ = A’ + B’ and (A + B)’ = A’ . B’

De Morgan’s theorems do heaviest lifting in exam questions because they let you push a NOT inside a bracket by swapping AND for OR. If a simplification feels stuck, applying De Morgan’s is usually way through.

Worked Example

Simplify A + A’B:

Distribute using the identity A + A’B = (A + A’)(A + B)
A + A’ = 1 by the complement law
1 . (A + B) = A + B by the identity law

So A + A’B simplifies to A + B, saving a gate in any circuit built from it. The calculator lays out these same steps for any expression you enter.

Truth Tables and Logic Gates

A truth table lists every combination of input values with resulting output, which makes it definitive way to prove two expressions are equivalent, identical tables mean identical logic. Two variables give 4 rows, three give 8 and each extra variable doubles count, which is exactly why calculator becomes useful beyond three variables.

Each Boolean operator corresponds to a physical logic gate, AND, OR and NOT are fundamentals, while NAND, NOR, XOR and XNOR combine them. Simplifying an expression before building it means fewer gates, in circuit design fewer gates means lower cost, less power and faster switching. That link between the algebra and the hardware is the whole point of the subject.

Boolean Algebra for GCSE and A Level Computer Science

Boolean logic sits in every UK Computer Science specification and notation varies by level. GCSE papers mostly use written operators and truth tables, while A Level introduces algebraic notation with overbars or apostrophes for NOT, plus simplification using the laws above and Karnaugh maps on some specifications.

The step by step output here matches working examiners expect, name the law, apply it, write new expression. Practise on paper first and use calculator to check each line, because spotting exactly where your working diverges is the fastest way to fix a misunderstood law.