# Set Operations

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.Binary operations may or may not have certain properties when applied to sets. We will define

$\ast$

as the operation.The symbol

$\forall$

means *, an example being***for all**$\forall a \in \N, 2a = a + a$

(for all $a$

in the natural numbers, $2a$

is equal to $a + a$

).The symbol

$\exists$

means **there exists**.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 $a \times b \in \N$

.$\forall a,b \in \N, a \ast b \in \N$

If changing the order of the element has no effect on the result, the operation is

**commutative**.$\forall a,b \in \N, a \ast b = b \ast a$

If the order of operations doesn't matter, the operation is

**associative**.$\forall a,b,c \in \N, (a \ast b) \ast c = a \ast (b \ast c)$

Last modified 1yr ago