What is a Truth Table?
A truth table is a mathematical tool used in propositional logic to determine all possible outcomes of a logical expression based on the truth values of its variables. Each row represents one combination of true (T) and false (F) values for the variables, and the final column shows the result of evaluating the full expression.
Truth tables are fundamental in digital electronics, computer science, philosophy, and mathematics. They are used to verify logical equivalences, design logic circuits, test software conditions, and understand how complex boolean expressions behave across every possible scenario.
For example, the expression "A AND B" is only true when both A and B are true. With two variables, there are four possible combinations (2^2 = 4 rows). With three variables, you get eight rows (2^3 = 8). This tool supports up to four variables, generating tables with up to 16 rows.
Supported Operators
This tool supports all standard propositional logic operators. You can use word-form operators or their symbolic equivalents:
**Conjunction (AND)** - Keywords: \`AND\`, \`&&\` - True only when both operands are true. - Example: \`A AND B\`
**Disjunction (OR)** - Keywords: \`OR\`, \`||\` - True when at least one operand is true. - Example: \`A OR B\`
**Negation (NOT)** - Keywords: \`NOT\`, \`!\` - Inverts the truth value of an operand. - Example: \`NOT A\`, \`!A\`
**Exclusive OR (XOR)** - Keywords: \`XOR\`, \`^\` - True when exactly one operand is true, but not both. - Example: \`A XOR B\`
**Implication (IMPL / ->)** - Keywords: \`IMPL\`, \`->\` - False only when the left side is true and the right side is false. Equivalent to "if A then B." - Example: \`A IMPL B\`, \`A -> B\`
**Biconditional (BICON / <->)** - Keywords: \`BICON\`, \`<->\` - True when both operands have the same truth value. Equivalent to "A if and only if B." - Example: \`A BICON B\`, \`A <-> B\`
**Grouping with Parentheses** - Use parentheses to control evaluation order. - Example: \`(A OR B) AND NOT C\`
How to Use
1. Type a logical expression into the input field, or click one of the preset examples to load it. 2. Use single uppercase letters (A, B, C, D) as your variable names. The tool detects them automatically. 3. Click "Generate Table" to compute all combinations. 4. Read the truth table: each row is one combination of variable values. The rightmost column shows whether the full expression evaluates to T (true) or F (false). 5. Check the summary below the table to see how many rows evaluate to true versus false. 6. Use the "Copy Table" button to copy the table as plain text for use in documents or reports.
Expression Examples
Here are some common logical expressions you can try:
- \`A AND B\` — conjunction: true only when both are true - \`A OR B\` — disjunction: true when either is true - \`NOT A\` — negation: flips the value - \`A XOR B\` — exclusive or: true when exactly one is true - \`A IMPL B\` — implication: false only when A is true and B is false - \`A BICON B\` — biconditional: true when A and B match - \`(A OR B) AND NOT C\` — compound expression with three variables - \`(A AND B) OR (NOT A AND C)\` — a more complex three-variable expression - \`A IMPL (B AND C)\` — implication with a compound consequent
Expressions can use any combination of these operators and any number of variables up to four (A, B, C, D).
FAQs
Q: What operators are supported? A: AND, OR, NOT, XOR, IMPL (implication), BICON (biconditional), and their symbolic equivalents (&&, ||, !, ^, ->, <->).
Q: How many variables can I use? A: Up to 4 variables (A, B, C, D) for manageable table sizes (up to 16 rows).
Q: Are variable names case-sensitive? A: Yes. Variables must be single uppercase letters: A, B, C, or D. Lowercase letters are not treated as variables.
Q: What does "implication" mean in logic? A: "A IMPL B" means "if A then B." It is false only when A is true and B is false. In all other cases it is true. This often surprises beginners because the statement is considered vacuously true when A is false.
Q: What is a biconditional? A: "A BICON B" means "A if and only if B." It is true when A and B have the same value (both true or both false) and false when they differ.
Q: Can I use this to check logical equivalences? A: Yes. If two expressions have identical result columns across all rows, they are logically equivalent. You can test one at a time and compare manually.
Q: Is my data stored or sent to a server? A: No. All calculations happen entirely in your browser. Nothing is transmitted to any server.
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