Truth Table Calculator
A truth table is one of the simplest and most powerful tools in logic. It is a compact grid that lists every possible combination of truth values for a set of statements, alongside the result of a logical operation applied to them. With just two inputs there are only four combinations to consider, which means the entire behavior of an operator like AND, OR, or XOR fits on a tiny table.
Because truth tables leave nothing to interpretation, they show up everywhere from philosophy classrooms to processor design. A Truth Table Calculator removes the tedium of filling in each row by hand, generating the complete table for any of the seven basic operators instantly. This guide explains what truth tables are, how each operator works, how to read and build tables yourself, and where they matter in circuits and programming.
What a Truth Table Is
In propositional logic, every statement is either true or false, with no middle ground. A proposition such as “it is raining” has a truth value that depends on the world, and compound statements combine simple ones with operators like AND, OR, and NOT. A truth table enumerates every possible assignment of truth values to the variables involved, then shows the resulting value of the compound statement for each assignment.
With one variable there are two rows. With two variables there are four rows: 00, 01, 10, and 11, where 1 stands for true and 0 stands for false. With three variables there are eight rows, and in general n variables produce 2^n rows. The table grows quickly, which is why most hand-built examples stick to two variables, the case this calculator handles.
The real value of the table is that it turns meaning into mechanics. Instead of debating what “A or B” ought to mean, you can look at the row where A is false and B is false and see the defined result. Every digital device rests on tables like these, evaluated billions of times per second inside silicon.
Propositions and Truth Values
A proposition is a declarative sentence that is unambiguously true or false. “Two plus two equals four” is a true proposition, “Paris is the capital of Italy” is a false one, and “Close the door” is not a proposition at all because it is a command rather than a claim. Logic manipulates these truth values the way arithmetic manipulates numbers.
- True is written as 1 or T, and it is the value of every correct statement.
- False is written as 0 or F, and it is the value of every incorrect statement.
- NOT flips a value, turning 1 into 0 and 0 into 1.
- Compound propositions combine values with operators such as AND, OR, and XOR.
Two-valued logic may sound restrictive, yet it captures the decisions that hardware and software actually make. A sensor reading is either above threshold or not, and a password either matches or fails. Each of these is a proposition with a true or false value, and operators combine them into larger decisions.
The Seven Basic Boolean Operators
Seven operators cover nearly everything done with two binary variables. The calculator above generates the table for each one. Here is what every row of those tables means.
AND: True Only When Both Inputs Are True
Conjunction, written A ∧ B, is the strictest operator: the result is 1 only in the single row where A equals 1 and B equals 1. In every other row the result is 0. This matches the everyday sense of “and” in a requirement like “bring a ticket and an ID”.
OR: True When At Least One Input Is True
Disjunction, written A ∨ B, is inclusive: the result is 1 whenever A is 1, B is 1, or both are 1. Only the row where both inputs are 0 produces 0. Note that logical OR differs from the exclusive “either but not both” of menu choices, which is the job of XOR.
XOR: True When the Inputs Differ
Exclusive OR, written A ⊕ B, is true exactly when the two inputs disagree, giving 1 for the rows 01 and 10 and 0 for 00 and 11. It is the operator behind parity checks and the addition of single bits in adder circuits.
NAND: False Only When Both Inputs Are True
NAND is the negation of AND, written A ↑ B, and its table is the AND table with every result flipped: three rows of 1 and a single 0 where both inputs are 1. It earns special attention because any logic circuit can be built using only NAND gates, a property called functional completeness.
NOR: True Only When Both Inputs Are False
NOR, written A ↓ B, negates OR: the result is 1 only in the row where both inputs are 0, and 0 everywhere else. Like NAND, it is functionally complete on its own, and NOR gates were the workhorse of early computing hardware such as the Apollo Guidance Computer.
Implication: False Only When a True Statement Implies a False One
Material implication, written A → B, surprises newcomers: it is false in exactly one row, where A is 1 and B is 0, and true in the other three. The idea is that a promise “if A then B” is broken only when A holds and B fails. When A is false the promise makes no claim and stays true.
Equivalence: True When Both Inputs Match
The biconditional, written A ↔ B, is true when A and B agree, giving 1 for rows 00 and 11 and 0 for rows 01 and 10. It expresses “A if and only if B”, used wherever two conditions must stand or fall together.
How to Read a Truth Table
Reading a table is a matter of locating the row that matches your inputs and reading across to the result column. The columns are conventional: input variables first, the result last.
- Identify the column headings so you know which variable is A and which is B.
- Find the row whose input values match your situation, for example A = 1 and B = 0.
- Move across that row to the result column to read the output value.
- Repeat for each row when you want the full behavior rather than a single case.
Try it on the calculator above: select Implication, locate the row for A = 1 and B = 0, and confirm the result is 0. Then compare with Equivalence, where the true rows are the two where the inputs agree.
How to Build a Truth Table from an Expression
Building a table by hand is a reliable skill that also teaches how expressions are evaluated. The method works for any combination of operators once the basic tables are known.
- List the variables and write every input combination, counting in binary: 00, 01, 10, 11.
- Break the expression into its smallest pieces, evaluating inner operations first.
- Add a column for each intermediate result, filling it using the operator tables.
- Combine the intermediate columns into the final column with the outermost operator.
For example, to tabulate (A ∧ B) ∨ ¬A, you would first compute the AND column, then the NOT A column, then OR the two together row by row. The calculator automates the single-operator case, a fast way to check each column before assembling the final result.
De Morgan’s Laws
Two of the most useful identities in logic are De Morgan’s laws, which describe how NOT distributes over AND and OR. The negation of a conjunction equals the disjunction of the negations, and the negation of a disjunction equals the conjunction of the negations. In symbols, ¬(A ∧ B) is equivalent to ¬A ∨ ¬B, and ¬(A ∨ B) is equivalent to ¬A ∧ ¬B.
You can verify either law with truth tables by building the table for the left side and the table for the right side and confirming that the final columns match row for row. This is exactly why NAND is defined as NOT-AND and NOR as NOT-OR: De Morgan’s laws show how each can be rewritten in terms of the other operators.
Truth Tables in Digital Circuits
Every logic gate in a processor is a physical truth table. An AND gate outputs a high voltage only when both inputs are high, which is precisely the AND table with 1 and 0 standing for voltage levels. Chip designers specify, test, and debug gates against these tables before a single transistor is laid out.
The universality of NAND and NOR explains their dominance in hardware: since either gate alone can reproduce every other gate, manufacturers can build entire circuits from one repeated component, simplifying fabrication. Adders, memory cells, and multiplexers all reduce to networks of gates whose combined truth table defines the behavior of the whole circuit.
- AND, OR, and NOT gates implement the three classic operations directly.
- Half-adders use XOR for the sum bit and AND for the carry bit.
- Flip-flops built from NAND or NOR gates store single bits of memory.
Truth Tables in Programming
Programmers meet truth tables every time they write a conditional. An expression like isAdmin AND hasPermission follows the AND table exactly, granting access only when both are true. Short-circuit evaluation even mirrors the table’s structure: once one input of an AND is false, the result is decided without checking the rest.
XOR appears in toggling, checksums, and swap algorithms, while implication shows up in validation rules of the form “if a field is present then it must be valid”. Database queries use the same logic, since WHERE clauses combine conditions with AND, OR, and NOT. Understanding the tables helps predict which rows a query will return and which branches a program will take.
Worked Examples
These examples connect the calculator output to concrete reasoning. Generate each table above and follow along.
Access control. Suppose entry requires a badge (A) and a PIN (B). The AND table shows access granted only in the row 11. If either factor is missing, rows 01, 10, and 00 all give 0, so the door stays locked. This is why two-factor authentication pairs two independent checks with AND rather than OR.
Free shipping rule. An order ships free if it is over 50 dollars (A) or the customer is a member (B). The OR table grants free shipping in rows 01, 10, and 11, denying it only in row 00. Changing the rule to “over 50 dollars and a member” would flip the table to AND, a one-word change with a four-row consequence.
Warranty promise. Consider the statement “if the product is registered (A) then the warranty is honored (B)”. The implication table is true in every row except 10, where registration happened but the warranty was denied, the only broken promise. Equivalence then checks whether registration status and warranty status always agree, which is the stricter guarantee a customer actually wants.
Conclusion
Truth tables compress the entire meaning of a logical operator into four rows. They define AND, OR, XOR, NAND, NOR, implication, and equivalence without ambiguity, they justify De Morgan’s laws by direct comparison, and they serve as the blueprint for the gates inside every computer. Whatever the scale of the system, the reasoning starts with these small grids.
Use the Truth Table Calculator above to generate any of the seven tables on demand, then apply the reading and building techniques from this guide to tackle larger expressions with confidence.
Frequently Asked Questions
What is a truth table used for?
A truth table defines the exact behavior of a logical operator by listing the result for every possible combination of inputs. It is used to learn logic, verify equivalences, design digital circuits, and predict how compound conditions behave in programs.
How many rows does a two-variable truth table have?
A two-variable truth table has four rows, one for each input pair: 00, 01, 10, and 11. In general, a table with n variables has 2 to the power n rows, so three variables need eight rows.
What is the difference between AND and OR?
AND is true only when both inputs are true, giving 1 in the row 11. OR is true when at least one input is true, giving 1 in rows 01, 10, and 11. Both are false in row 00.
What does XOR mean in logic?
XOR stands for exclusive or and is true exactly when the two inputs differ. Its table gives 1 for rows 01 and 10 and 0 for rows 00 and 11. It models “one or the other but not both”, and it powers parity checks and binary addition.
Why is NAND called a universal gate?
NAND is called universal because every other logic gate can be built from NAND gates alone, a property known as functional completeness. The same holds for NOR. This lets chip makers fabricate complex circuits from a single repeated gate type.
What is the difference between NOR and NAND?
NOR is the negation of OR and is true only when both inputs are false, with its single 1 in row 00. NAND is the negation of AND and is false only when both inputs are true, with its single 0 in row 11.
When is an implication false?
A material implication A implies B is false in exactly one case: when A is true and B is false, the row 10. In every other row, 00, 01, and 11, the implication is true. A false antecedent never breaks the implication.
What does equivalence mean in a truth table?
Equivalence, written A if and only if B, is true when the two inputs match. Its table gives 1 for rows 00 and 11 and 0 for rows 01 and 10. It is the logical test for whether two statements always agree.
How do you read a truth table with three variables?
A three-variable table has eight rows covering 000 through 111. Read it the same way as a two-variable table: find the row matching your three input values, then read across to the result column.
What are De Morgan’s laws in simple terms?
De Morgan’s laws say that negating a combination flips the operator: NOT (A AND B) equals (NOT A) OR (NOT B), and NOT (A OR B) equals (NOT A) AND (NOT B). They let you push negations inward, which simplifies circuits and conditions.
Can a truth table prove two expressions equivalent?
Yes. Build the table for each expression over the same variables and compare the final columns row by row. If every row matches, the expressions are logically equivalent. This method is exhaustive and leaves no room for doubt.
How are truth tables used in computer programming?
Every boolean expression in code follows a truth table: if statements combine conditions with AND, OR, and NOT, database queries filter rows the same way, and bitwise operators apply the tables to each bit. Understanding the tables helps predict exactly when a condition triggers.
What is the truth table for NOT?
NOT is a unary operator with a single input and two rows: an input of 0 gives 1 and an input of 1 gives 0. It simply flips the truth value. Combined with AND and OR, NOT completes the classic trio of boolean operations.
How do you build a truth table for a complex expression?
List all input combinations in binary order, then evaluate the expression piece by piece, adding a column for each intermediate result. Combine those columns with the outermost operator last. The calculator above can verify each single-operator column as you build.
Where can I practice truth tables for free?
The Truth Table Calculator at the top of this page generates all seven two-variable tables instantly and is free to use. Textbooks on discrete mathematics and online logic courses also provide unlimited practice.