How this instrument works
A truth table lists every possible outcome of a logical operation for every combination of yes/no inputs. This page computes three of the most common operations at once for a single chosen pair: AND (affirmative only when both inputs hold), OR (affirmative whenever either input holds), and XOR (affirmative when the pair disagrees with each other).
Each operation is computed here using a plain arithmetic identity rather than a dedicated logic gate, a trick that works precisely because the inputs are restricted to exactly two values: AND is simply the product p×q, OR is p+q−p×q (adding both and correcting for double-counting the shared case), and XOR is p+q−2×p×q (that same correction doubled, so agreeing inputs cancel each other out).
These three operations are the basic building blocks digital logic circuits are built from — every more complex logical expression, in principle, can be constructed by combining AND, OR, and a NOT operation (flipping the input to its opposite) in the right arrangement.
- Enter p as either 0 (false) or 1 (true).
- Enter q the same way.
- Read AND, OR, and XOR — all three results computed at once for that single pair of inputs.
- Try all four combinations of p and q (0,0 / 0,1 / 1,0 / 1,1) to build out the complete truth table by hand.
Worked example — p=1, q=0
With p on and q off (p=1, q=0): AND multiplies out to 1×0=0 (both inputs must hold for AND to fire, and here only one does), OR comes out to 1+0−0=1 (at least one input holds), and XOR also comes out affirmative, since the pair disagrees with each other.
With both p and q switched on (p=1, q=1, matching each other): AND fires, OR fires, but XOR falls silent, dropping to 1+1−2=0 — it specifically flags DISAGREEMENT between the pair, and here there isn't any. With both switched off (p=0, q=0, also matching): every one of the three results falls silent, since neither input ever holds to begin with.
Questions
What is a truth table?
A complete listing of a logical operation's output for every possible combination of 0/1 inputs — for two inputs, that means exactly four rows: 0-0, 0-1, 1-0, and 1-1.
How can arithmetic stand in for logic gates?
When inputs are restricted to exactly 0 and 1, ordinary multiplication and addition reproduce the logic gates exactly — multiplication behaves like AND, and the right combination of addition and multiplication reproduces OR and XOR, since those are the only values 0 and 1 can ever take.
What is XOR, and how is it different from OR?
XOR (exclusive or) holds only when the two inputs DISAGREE — one at 1 and the other at 0; ordinary OR holds whenever AT LEAST ONE input equals 1, including the case where both do, which is exactly the case where XOR and OR give different answers.
What is AND used for?
Testing whether two conditions BOTH hold at once — it returns 1 only in the single case where both inputs already equal 1, making it the strictest of the three operations computed here.
Can these three operations build any logical expression?
In principle, yes — AND, OR, and a NOT operation (inverting 0 and 1) together form a functionally complete set, meaning any logical expression at all can be built by combining them in the right arrangement.