Often in maths, we take two numbers
and perform an operation on the two. Unary operations only involve one number, for example calculating
. Binary operations take two numbers, for example
. The latter are especially interesting when it comes to Group Theory in Cryptography.
Binary operations may or may not have certain properties when applied to sets. We will define
as the operation.
means for all, an example being
in the natural numbers,
is equal to
means there exists.
If the result of
is within the original set, then the operation is closed. An example is multiplication in
, as for any pair
we can see that
If changing the order of the element has no effect on the result, the operation is commutative.
If the order of operations doesn't matter, the operation is associative.