XOR, the odd one out

One or the other, but not both

The OR lesson warned that the gate is not the "or" of everyday speech: OR is happy when both inputs are 1. There is a gate for the everyday meaning too. It outputs 1 when one input or the other is 1, but not when both are, and not when neither is. It is called XOR, short for "exclusive or", because it excludes the case where both are on.

The cleanest way to remember it is not about "or" at all: XOR outputs 1 when its two inputs are different, and 0 when they are the same. Look at the table. 00 and 11, the two rows where the inputs match, give 0. 01 and 10, where they differ, give 1. The symbol is the OR shape with a second curved line drawn behind its back.

Adding without the carry

Here is why XOR earns a place among the seven. Add two single bits and there are four cases: 0 + 0 = 0, 0 + 1 = 1, 1 + 0 = 1, and 1 + 1 = 10, which is binary for two: a 0 in this column and a 1 carried to the next. Now look at just the digit that stays in this column: 0, 1, 1, 0. That is the XOR table. And look at the carry: 0, 0, 0, 1. That is the AND table.

So one XOR gate and one AND gate, side by side on the same two inputs, add two bits. The XOR gives the sum digit and the AND gives the carry. That little circuit has a name, the half adder, and it is the first thing built in the stage on arithmetic. Everything a computer does with numbers starts here.

A switch that flips

Set one input of an XOR gate to a fixed value and something useful happens to the other. Hold b at 0 and the output simply copies a: 0 gives 0, 1 gives 1. Hold b at 1 and the output is the opposite of a: 0 gives 1, 1 gives 0. So an XOR gate is a NOT gate that can be switched on and off by its second input. Engineers call this a controlled inverter, and it is how a circuit is told "use this number as it is" or "use its opposite" with a single control wire.

Counting 1s: parity

Chain XOR gates together and the pattern continues. Three inputs XORed give 1 when an odd number of them are 1 and 0 when an even number are. Click through the widget below: the output is 1 for exactly one input on or all three on, and 0 for none or two. This is called the parity of the inputs, and it is a cheap way to check that a set of bits arrived intact. Send the bits plus one extra bit that makes the count of 1s even; if the receiver counts an odd number, something was flipped on the way.

What to remember

• XOR outputs 1 when its two inputs are different and 0 when they are the same.
• It is the everyday "or": one or the other, but not both.
• XOR is the sum digit when two bits are added; AND is the carry. Together they make a half adder.
• XOR with a 1 flips the other input; XOR with a 0 leaves it alone. That is a controlled inverter.
• A chain of XORs tells you whether the number of 1s is odd, which is called parity.