LearnTronics
Truth Tables
A truth table is just organized bookkeeping
A truth table lists every possible combination of input truth values and shows the resulting output.
Two inputs means four rows
With A and B, each can be False or True. That gives:
| A | B |
|---|---|
| F | F |
| F | T |
| T | F |
| T | T |
The order itself is not magic. The important part is that we include every combination once.
The basic operators together
| A | B | AND | OR | XOR |
|---|---|---|---|---|
| F | F | F | F | F |
| F | T | F | T | T |
| T | F | F | T | T |
| T | T | T | T | F |
Another way to think of it
A truth table is like testing a machine in every possible switch position. Put switch A one way and B one way; record the result. Change one switch; record the result. Continue until every combination has been tried.
Truth tables are not limited to syllogisms
They can describe any Boolean expression with a finite set of true/false inputs. With three inputs there are 2³ = 8 combinations. With four inputs there are 2⁴ = 16.
A little deeper
There are sixteen possible Boolean functions of two inputs because each function can choose either False or True for each of the four input rows: 2⁴ = 16. We do not need to memorize all sixteen. The useful skill is learning to build the table correctly from the expression.