Truth tables.

Step 1 of 6

A row for every combination

A gate is a rule, and a truth table writes the rule out in full: a row for every possible combination of the inputs, and on each row what comes out. A gate with two inputs has four rows, usually written in the order of the binary numbers, 00, 01, 10, 11:

A	B	A AND B
$0$⁠	$0$⁠	$0$⁠
$0$⁠	$1$⁠	$0$⁠
$1$⁠	$0$⁠	$0$⁠
$1$⁠	$1$⁠	$1$⁠

The four settings of Input bits are exactly the four rows.

Step 2 of 6

The OR column

To fill in a column, go through the rows one by one. On the board you can do it by experiment: set Input bits to each row in turn, and read the light.

Step 3 of 6

OR but not AND

The columns of OR and AND differ on two rows. Find one of them on the board.

Step 4 of 6

XOR

The rows you found define a fourth gate, XOR, “exclusive or”: 1 when exactly one input is 1. It is the “or” of everyday speech, tea or coffee but not both. XOR asks whether its two inputs differ, and in the next Lesson it will add numbers.

Step 5 of 6

Gates from gates

Gates can be joined: the output of one goes into another. That is how you write an expression like

$(A \text{ OR } B) \text{ AND NOT } (A \text{ AND } B)$⁠

and fill in its table in stages, a working column for each gate. To work it out, start from the brackets, as in arithmetic.

Step 6 of 6

Every gate

A two-input gate is decided entirely by its output column, so there are only so many such gates. The wonder is that you do not need them all: AND, OR and NOT alone can build any of them, and NAND alone, $\text{NOT}(A \text{ AND } B)$⁠, can build all the rest. So a whole processor can be made from one kind of gate.