Truth Table Generator – Create Logic Truth Tables Online
Generate truth tables for any logical expression with our free online truth table generator. Supports AND, OR, NOT, XOR, NAND, NOR, and implication operators for any number of variables.
Use variables: A, B
How the Truth Table Generator Works
A truth table systematically lists all possible combinations of input values for a logical expression and shows the resulting output for each combination. This tool evaluates boolean expressions using the fundamental operators of propositional logic.
For n variables, there are 2ⁿ possible combinations. With 2 variables you get 4 rows, with 3 variables you get 8 rows, and with 4 variables you get 16 rows. The generator evaluates your expression for each row and displays T (true) or F (false) for the result.
Supported operators:
• AND (∧): True only when both inputs are true
• OR (∨): True when at least one input is true
• NOT (¬): Inverts the truth value
• XOR (⊕): True when exactly one input is true
• NAND: NOT AND (false only when both are true)
• NOR: NOT OR (true only when both are false)
• IMPLIES (→): False only when first is true and second is false
Example Logical Expressions
Basic AND Operation
Expression: A AND B
| A | B | A AND B |
|---|---|---|
| T | T | T |
| T | F | F |
| F | T | F |
| F | F | F |
De Morgan's Law
Expression: NOT (A AND B) = (NOT A) OR (NOT B)
Test both sides to verify they produce identical results:
Left: NOT (A AND B)
Right: (NOT A) OR (NOT B)
Both expressions are logically equivalent
Implication (Conditional)
Expression: A IMPLIES B
This is equivalent to: (NOT A) OR B
Only false when A=T and B=F
Think: "If it rains, then the ground is wet"
False only if it rains but ground stays dry
Complex Expression
Expression: (A AND B) OR (NOT C)
True when:
• Both A and B are true, OR
• C is false
• (or both conditions)
Quick Fact: Boolean Logic Origins
George Boole (1815-1864), an English mathematician with no formal university education, created Boolean algebra in his 1854 book "The Laws of Thought." He showed that logic could be expressed using mathematical symbols where variables take only two values: 0 (false) or 1 (true). Nearly 80 years later, Claude Shannon realized in his 1937 MIT master's thesis that Boolean algebra could describe electrical switching circuits—laying the foundation for all digital computers. Every processor, memory chip, and digital device today operates on principles Boole established.
Frequently Asked Questions
What is a truth table used for?
Truth tables are used to analyze logical expressions, verify logical equivalences, test argument validity, design digital circuits, and debug boolean conditions in programming. They provide a complete picture of how an expression behaves under all possible input conditions.
How do I know if two expressions are logically equivalent?
Two expressions are logically equivalent if their truth tables produce identical output columns for every row. For example, "NOT (A AND B)" and "(NOT A) OR (NOT B)" are equivalent (De Morgan's Law). Generate truth tables for both and compare the result columns.
What does "A IMPLIES B" really mean?
"A implies B" (written A → B) is false only when A is true and B is false. It's true in all other cases, including when A is false. This seems odd but makes sense: "If it's raining, then the ground is wet" isn't proven false on a sunny day—the statement only fails if it rains and the ground stays dry.
What's the difference between OR and XOR?
Regular OR (inclusive OR) is true when at least one input is true—including when both are true. XOR (exclusive OR) is true only when exactly one input is true, but false when both are true. XOR is like "either/or but not both."
How are truth tables used in computer science?
In programming, truth tables help design complex conditional statements, simplify boolean expressions, and debug logic errors. In hardware design, they specify the behavior of logic gates and digital circuits. Database queries and search filters also rely on boolean logic that can be analyzed with truth tables.
What is a tautology?
A tautology is an expression that's always true, regardless of input values. For example, "A OR (NOT A)" is a tautology—it's true whether A is true or false. In a truth table, a tautology shows all T values in the result column. The opposite is a contradiction (always false).
Can I use parentheses in my expressions?
Yes! Parentheses control the order of operations, just like in regular math. "(A AND B) OR C" is different from "A AND (B OR C)". Use parentheses to make your intended grouping clear, especially in complex expressions with multiple operators.
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