Enter a logical expression using variables P, Q, R, S, T and operators like AND, OR, NOT, XOR, IMPLIES, IFF. The calculator will generate a complete truth table for all possible truth value combinations.
Truth Table:
// Define logical operators and their properties for Shunting-Yard algorithm
var operators = {
'NOT': { precedence: 4, type: 'unary', associativity: 'right' },
'AND': { precedence: 3, type: 'binary', associativity: 'left' },
'XOR': { precedence: 2, type: 'binary', associativity: 'left' },
'OR': { precedence: 1, type: 'binary', associativity: 'left' },
'IMPLIES': { precedence: 0, type: 'binary', associativity: 'right' }, // P -> Q is !P || Q
'IFF': { precedence: 0, type: 'binary', associativity: 'left' } // P Q is (P == Q)
};
// Helper functions for logical operations
function logicalImplies(a, b) { return !a || b; }
function logicalIff(a, b) { return a == b; }
// Tokenizer function: Breaks the expression string into meaningful tokens
function tokenize(expression) {
expression = expression.toUpperCase().replace(/\s+/g, "); // Remove all whitespace and convert to uppercase
var tokens = [];
var i = 0;
// Regex to match variables (P,Q,R,S,T), operators, and parentheses
var regex = /(P|Q|R|S|T)|(NOT|AND|OR|XOR|IMPLIES|IFF)|(\(|\))/g;
var match;
while ((match = regex.exec(expression)) !== null) {
if (match.index > i) {
// If there's unparsed text between matches, it's an invalid character
throw new Error("Invalid character or sequence: " + expression.substring(i, match.index));
}
tokens.push(match[0]);
i = regex.lastIndex;
}
if (i < expression.length) {
// If there's unparsed text at the end, it's an invalid character
throw new Error("Invalid character or sequence at end: " + expression.substring(i));
}
return tokens;
}
// Shunting-Yard algorithm: Converts an infix expression to Reverse Polish Notation (RPN)
function shuntingYard(expression) {
var outputQueue = [];
var operatorStack = [];
var tokens = tokenize(expression);
for (var i = 0; i 0) {
var op2Token = operatorStack[operatorStack.length – 1];
if (op2Token === '(') break; // Stop if we hit an open parenthesis
var op2 = operators[op2Token];
// Compare precedence and associativity
if (op2 && ((op1.associativity === 'left' && op1.precedence <= op2.precedence) ||
(op1.associativity === 'right' && op1.precedence 0 && operatorStack[operatorStack.length – 1] !== '(') {
outputQueue.push(operatorStack.pop());
}
if (operatorStack.length === 0 || operatorStack[operatorStack.length – 1] !== '(') {
throw new Error("Mismatched parentheses.");
}
operatorStack.pop(); // Pop the '(' from the stack
}
// Unknown token
else {
throw new Error("Unknown token: " + token);
}
}
// Pop any remaining operators from the stack to the output queue
while (operatorStack.length > 0) {
var op = operatorStack.pop();
if (op === '(' || op === ')') {
throw new Error("Mismatched parentheses.");
}
outputQueue.push(op);
}
return outputQueue;
}
// Evaluate RPN expression: Calculates the truth value of an RPN expression
function evaluateRPN(rpnTokens, varValues) {
var stack = [];
for (var i = 0; i < rpnTokens.length; i++) {
var token = rpnTokens[i];
// If token is a variable, push its truth value to the stack
if (token === 'P' || token === 'Q' || token === 'R' || token === 'S' || token === 'T') {
stack.push(varValues[token]);
}
// If token is an operator, perform the operation
else if (token in operators) {
var op = operators[token];
if (op.type === 'binary') { // Binary operators (AND, OR, XOR, IMPLIES, IFF)
if (stack.length < 2) throw new Error("Syntax error: not enough operands for " + token);
var b = stack.pop();
var a = stack.pop();
if (token === 'AND') stack.push(a && b);
else if (token === 'OR') stack.push(a || b);
else if (token === 'XOR') stack.push(a != b); // Logical XOR (true if different)
else if (token === 'IMPLIES') stack.push(logicalImplies(a, b));
else if (token === 'IFF') stack.push(logicalIff(a, b));
} else if (op.type === 'unary') { // Unary operator (NOT)
if (stack.length < 1) throw new Error("Syntax error: not enough operands for " + token);
var a = stack.pop();
stack.push(!a);
}
}
// Unknown token in RPN
else {
throw new Error("Unknown token in RPN: " + token);
}
}
if (stack.length !== 1) throw new Error("Syntax error: too many operands or operators.");
return stack.pop();
}
// Main function to calculate and display the truth table
function calculateTruthTable() {
var expression = document.getElementById("logicalExpression").value;
var resultDiv = document.getElementById("truthTableContent");
resultDiv.innerHTML = ''; // Clear previous results
if (!expression.trim()) {
resultDiv.innerHTML = 'Please enter a logical expression.';
return;
}
var uniqueVars = [];
try {
// Identify unique variables (P, Q, R, S, T) present in the expression
var varRegex = /\b(P|Q|R|S|T)\b/g;
var matches = expression.toUpperCase().match(varRegex);
if (matches) {
uniqueVars = Array.from(new Set(matches)).sort(); // Sort for consistent column order
}
// Handle expressions with no variables (constant expressions)
if (uniqueVars.length === 0) {
var constantResult = evaluateRPN(shuntingYard(expression), {});
resultDiv.innerHTML = 'Expression: ' + expression + '' +
'Result: ' + (constantResult ? 'True' : 'False') + '';
return;
}
// Limit the number of variables to keep the table manageable
if (uniqueVars.length > 5) {
resultDiv.innerHTML = 'Too many variables. Please use a maximum of 5 variables (P, Q, R, S, T).';
return;
}
var numRows = Math.pow(2, uniqueVars.length);
var tableHTML = '
';
tableHTML += '
';
// Add headers for each variable
for (var i = 0; i < uniqueVars.length; i++) {
tableHTML += '
' + uniqueVars[i] + '
';
}
// Add header for the full expression
tableHTML += '
' + expression + '
';
tableHTML += '
';
var rpnExpression = shuntingYard(expression); // Convert to RPN once for efficiency
// Iterate through all possible truth value combinations
for (var i = 0; i < numRows; i++) {
var varValues = {};
tableHTML += '
';
for (var j = 0; j > (uniqueVars.length – 1 – j)) & 1) === 1;
varValues[uniqueVars[j]] = isTrue; // Store boolean value
tableHTML += '
' + (isTrue ? 'T' : 'F') + '
';
}
// Evaluate the expression for the current variable assignment
var result = evaluateRPN(rpnExpression, varValues);
tableHTML += '
A truth table is a mathematical table used in logic—specifically in Boolean algebra, Boolean functions, and propositional calculus—to compute the functional values of logical expressions on each of their functional arguments, that is, on each combination of truth values taken by their propositional variables.
Why Use a Truth Table?
Truth tables are fundamental tools for:
Verifying Logical Arguments: Determining if an argument is valid or if a statement is a tautology (always true), contradiction (always false), or contingency.
Simplifying Boolean Expressions: Identifying equivalent expressions, which is crucial in digital circuit design and computer programming.
Understanding Operator Behavior: Clearly illustrating how different logical operators (AND, OR, NOT, etc.) combine truth values.
Computer Science: Essential for designing logic gates, understanding conditional statements, and optimizing algorithms.
Key Logical Operators Explained
This calculator supports the following common logical operators:
NOT (Negation): Reverses the truth value of a statement. If P is True, NOT P is False. If P is False, NOT P is True.
AND (Conjunction): True only if all statements connected by AND are True. Otherwise, it's False. (e.g., P AND Q)
OR (Disjunction): True if at least one of the statements connected by OR is True. False only if all statements are False. (e.g., P OR Q)
XOR (Exclusive OR): True if exactly one of the statements is True, and the other is False. False if both are True or both are False. (e.g., P XOR Q)
IMPLIES (Conditional): Represented as P → Q. It is False only if P is True and Q is False. In all other cases, it is True. (e.g., P IMPLIES Q)
IFF (Biconditional): Represented as P ↔ Q. It is True if P and Q have the same truth value (both True or both False). Otherwise, it's False. (e.g., P IFF Q)
Parentheses () can be used to group expressions and control the order of operations.
How to Use the Truth Table Calculator
Enter Your Expression: Type your logical expression into the "Logical Expression" input field.
Use Supported Variables: You can use single uppercase letters P, Q, R, S, T as your propositional variables.
Use Supported Operators: Input operators as NOT, AND, OR, XOR, IMPLIES, and IFF.
Group with Parentheses: Use () to define the order of operations, just like in algebra.
Generate Table: Click the "Generate Truth Table" button. The calculator will display a table showing the truth value for each variable combination and the final result of your expression.
Examples of Logical Expressions:
P AND Q
NOT (P OR Q)
P IMPLIES Q
(P AND Q) IFF R
(P XOR Q) OR (NOT R)
The calculator will automatically identify the variables in your expression and generate a table with 2n rows, where 'n' is the number of unique variables (up to 5 variables supported).