Truth Tables
A truth table lists every possible input combination and the output the circuit produces, defining a Boolean function exactly.
What a truth table is
A truth table is an exhaustive lookup. For a function of n inputs there are 2 to the n rows, one per combination. The table pairs each combination with the output, leaving nothing to interpretation.
Counting the rows
One input gives 2 rows, two inputs give 4, three give 8, four give 16. Rows are conventionally listed in binary counting order, so the input columns read like a binary counter from all zeros to all ones.
Example: two-input AND
| A | B | A AND B |
|---|---|---|
| 0 | 0 | 0 |
| 0 | 1 | 0 |
| 1 | 0 | 0 |
| 1 | 1 | 1 |
Why they matter
- They define a function without ambiguity, independent of how it is built.
- Two circuits are logically equal exactly when their truth tables match.
- They are the starting point for minimization with a Karnaugh map or algebra.
- They map directly to a sum-of-products expression by reading off the rows that output 1.
From table to expression
Each row that outputs 1 contributes one product term, called a minterm, in which every input appears either plain or complemented. ORing these minterms reproduces the function. The dual reading, using the rows that output 0, gives a product-of-sums form.
Because a truth table grows exponentially with input count, large functions are usually specified more compactly, but for a handful of inputs the table remains the clearest and most reliable description.