Set Operations

Operation Types

Often in maths, we take two numbers
aa
and
bb
and perform an operation on the two. Unary operations only involve one number, for example calculating
a−1a^{-1}
. Binary operations take two numbers, for example
a×ba \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
∀a∈N,2a=a+a\forall a \in \N, 2a = a + a
(for all
aa
in the natural numbers,
2a2a
is equal to
a+aa + a
).
The symbol
∃\exists
means there exists.

Closure

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

Associativity

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