Set Operations

Operation Types
Often in maths, we take two numbers aaa
 and bbb
 and perform an operation on the two. Unary operations only involve one number, for example calculating a−1a^{-1}a−1
. Binary operations take two numbers, for example a×ba \times ba×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∀a∈N,2a=a+a
 (for all aaa
 in the natural numbers, 2a2a2a
 is equal to a+aa + aa+a
).
The symbol ∃\exists∃
 means there exists.
Closure
If the result of a∗ba \ast ba∗b
 is within the original set, then the operation is closed. An example is multiplication in N\NN
, as for any pair a,ba,ba,b
 we can see that a×b∈Na \times b \in \Na×b∈N
.
∀a,b∈N,a∗b∈N\forall a,b \in \N, a \ast b \in \N∀a,b∈N,a∗b∈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∀a,b∈N,a∗b=b∗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)∀a,b,c∈N,(a∗b)∗c=a∗(b∗c)
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