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A truth table lays out every possible combination of true/false values for a boolean expression's variables, alongside the resulting output for each one. This tool builds the complete table for expressions using AND, OR, NOT, XOR, and IMPLIES over up to 4 variables (A-D).
How it works
The expression is parsed respecting standard operator precedence (NOT binds tightest, then AND, then XOR, then OR, then IMPLIES), and every one of the 2ⁿ true/false combinations for its variables is substituted in and evaluated.
- Enter boolean expression (use A-D, AND, OR, NOT, XOR, IMPLIES).
- Click Calculate to see your results.
Examples
A simple AND
A AND B is true only when both A and B are true — 1 of the 4 rows is true.
An XOR expression
A XOR B is true exactly when A and B differ — 2 of the 4 rows are true.
Who should use it
- Discrete mathematics and logic coursework.
- Computer science coursework on boolean algebra and digital logic.
Industry applications
- Computer science and digital logic design education
- Discrete mathematics
Advantages
- Supports all 5 standard logical connectives plus parentheses.
- Automatically classifies the result as a tautology, contradiction, or contingency.
Limitations
- Limited to 4 variables (A-D) to keep the truth table a manageable size (16 rows maximum).
Common mistakes to avoid
- Forgetting that IMPLIES (A → B) is only false when A is true and B is false — every other combination makes it true, which surprises many students at first.
Best practices
- Use parentheses liberally to make operator precedence explicit rather than relying on the default AND/OR/XOR/IMPLIES ordering.
Tips
- A contradiction (always false) and its negation (a tautology, always true) are logically linked — negating a known contradiction is a quick way to build a guaranteed tautology.