# Set Operations

## Operation Types

Often in maths, we take two numbers
$a$
and
$b$
and perform an operation on the two. Unary operations only involve one number, for example calculating
$a^{-1}$
. Binary operations take two numbers, for example
$a \times b$
. The latter are especially interesting when it comes to Group Theory in Cryptography.

## Properties of Binary Operations

Binary operations may or may not have certain properties when applied to sets. We will define
$\ast$
as the operation.
The symbol
$\forall$
means for all, an example being
$\forall a \in \N, 2a = a + a$
(for all
$a$
in the natural numbers,
$2a$
is equal to
$a + a$
).
The symbol
$\exists$
means there exists.

### Closure

If the result of
$a \ast b$
is within the original set, then the operation is closed. An example is multiplication in
$\N$
, as for any pair
$a,b$
we can see that
$a \times b \in \N$
.
$\forall a,b \in \N, a \ast b \in \N$

### Commutativity

If changing the order of the element has no effect on the result, the operation is commutative.
$\forall a,b \in \N, a \ast b = b \ast a$

### Associativity

If the order of operations doesn't matter, the operation is associative.
$\forall a,b,c \in \N, (a \ast b) \ast c = a \ast (b \ast c)$