Boolean algebra simplifier — minimal SOP and POS

How to use the Boolean algebra simplifier

1. Type the expression in Boolean expression, for example A'B + AB + AB'.
2. Keep Single-letter variables (AB means A AND B) ticked for textbook-style input, or untick it for longer variable names.
3. Choose an Output notation: Algebra (AB’ + C), Logic (A ∧ ¬B ∨ C), Words (A AND NOT B OR C) or Code (A && !B || C).
4. Read the Minimal sum of products and Minimal product of sums, and check whether the expression is a tautology, a contradiction or contingent.
5. Use Parsed as to confirm the tool read your brackets the way you meant, and download the truth table if needed.

What it does and when to use it

Simplifying Boolean expressions is a core skill in digital electronics, computer science and discrete maths. A shorter expression needs fewer logic gates and is easier to read in code. Students use Karnaugh maps or algebra laws by hand; this tool gives the answer to check against, plus the minterm list that a K-map starts from.

To compare several expressions in one table, use the truth table generator. For the same logic on actual bits, such as masks, try the binary calculator. The set theory calculator covers the matching laws for unions and intersections.

How it works

1. The expression is parsed with this order, from strongest to weakest: NOT, AND (including side-by-side letters), XOR/XNOR, OR/NOR, IMPLIES, IFF.
2. It is evaluated for every combination of inputs to build the truth table and the minterm list.
3. The Quine–McCluskey method repeatedly merges minterms that differ in one bit, which gives the prime implicants.
4. Essential prime implicants are taken first, then the smallest set that covers the remaining minterms is chosen. That is the minimal sum of products.
5. The same process on the 0 rows gives the minimal product of sums.

Step by step for A’B + AB + AB’:

1. Minterms: A’B is row 01 = 1, AB’ is row 10 = 2, AB is row 11 = 3.
2. Merge 01 and 11: they differ only in A, giving -1, which is B.
3. Merge 10 and 11: they differ only in B, giving 1-, which is A.
4. A and B together cover rows 1, 2 and 3, so the answer is A + B.

By algebra: AB + AB’ = A(B + B’) = A, then A + A’B = A + B.

Worked examples

Three terms to two letters. A'B + AB + AB' gives Minimal sum of products: A + B, with minterms 1, 2, 3 and maxterm 0.

Majority function. AB + BC + AC is already minimal as a sum of products (AB + AC + BC). Its product-of-sums form is (A + B)(A + C)(B + C).

Tautology. A + A' is always true, so the tool reports a tautology (always 1).

XOR. With single letters off, A XOR B becomes A'B + AB', or (A + B)(A’ + B’) as a product of sums.

Limits and tips

• Put brackets around anything you are unsure of; implicit AND binds tighter than OR, so AB + C means (A AND B) OR C.
• Don’t-care conditions cannot be entered.
• All processing happens in your browser.