anninmathbbN

AnninmathbbN is a mathematical construct defined as a subset of the natural numbers characterized by a parity condition on binary representations. Formally, anninmathbbN = { n ∈ N : the binary expansion of n contains an even number of 1s }. The element 0 belongs to this set since its binary form has zero 1s, an even count.

Notation and interpretation

Often denoted simply as A or as anninmathbbN in informal usage, this set provides a straightforward example of a parity filter on the nonnegative integers. The property can be expressed via the Hamming weight function weight(n), which counts the number of 1s in the binary expansion: n ∈ anninmathbbN if weight(n) ≡ 0 (mod 2).

Basic properties

- A is infinite and has natural density 1/2 in the natural numbers, since half the integers have an even number of 1s in their binary form.

- Under the bitwise XOR operation, the set anninmathbbN forms a subgroup of the corresponding algebraic structure, as weight(n ⊕ m) ≡ weight(n) + weight(m) (mod 2).

- Under ordinary addition, anninmathbbN is not closed; adding two numbers with even weight does not guarantee an even-weight result.

Examples

0, 3 (11), 5 (101), 6 (110), 9 (1001), 10 (1010), 12 (1100), 15 (1111), 17 (10001) belong to anninmathbbN, while 1 (1), 2 (10), 4 (100), 7 (111) do not.

Generalizations and related concepts

Variants include anninN_k, the set of n with weight(n) ≡ 0 (mod k), and other base representations with analogous parity criteria. The concept relates to coding theory (even-weight codes) and to character theory on the XOR group of binary vectors.