Whenever I see the definition of a group, it usually explicitly requires closure under its binary operation. An example, from the Wikipedia article:
A group (G, *) is a set G with a binary operation * that satisfies the following four axioms:
- Closure: For all a, b in G, the result of a * b is also in G.
- Associativity: For all a, b and c in G, (a * b) * c = a * (b * c).
- Identity element: There exists an element e in G such that for all a in G, e * a = a * e = a.
- Inverse element: For each a in G, there exists an element b in G such that a * b = b * a = e, where e is an identity element.
To its credit, Wikipedia goes on the mention the thing that I'm complaining about. Closure is still included as an axiom, however. Other perpetrators don't even go that far.
The hell am I talking about?
A binary operation θ is a mapping from the set A × A to the set A, where × stands for the Cartesian product.
If we were to write θ as a function, it'd look something like this:
θ : (A × A) → A
So, basically, if b,c ∈A, then ∃d∈A such that θ(b,c) = d. Equivalently, we could say that b θ c = d, for some d in A. The only thing that you should really care about for this rant is that when you apply the binary operation to two elements of A, the result is another element of A. That is, A is closed under θ.
The definition of a binary operation implies closure. Groups are almost always (correctly) defined as possessing a binary operation; so, to say that a group is closed under its binary operation is redundant.