Question

Under which operation is the set {0, 1} closed?

Answer 1

Multiplication.

$0\times0=0$
$0\times1=0$
$1\times0=0$
$1\times1=1$

All the possible products are contained in the set, so the set is closed under multiplication.

This isn't true for any of the other three standard arithmetic operations (addition, subtraction, division), and it's easy to see why.

One caveat about addition, though. The set *is* closed under addition modulo 2. Any even number $x$ is equivalent to 0 modulo 2, i.e. $x\equiv 0\mod2$, while any odd number $y$ is equivalent to 1 modulo 2, i.e. $y\equiv1\mod2$.

When you add two even numbers together, you get another even number.
When you add two odd numbers together, you get an even number.
When you add an even number to an odd number, you get an odd number.

So if you represent every even number by 0 and every odd number by 1, you get the following results:

$0+0=0$ (even + even = even)
$0+1=1$ (even + odd = odd)
$1+0=1$ (odd + even = odd)
$1+1=0$ (odd + odd = even)

The same also goes for multiplication modulo 2.