Parityequality

Parityequality is a mathematical relation on the integers that holds between two integers when they have the same parity, meaning both are even or both are odd. While not a standardized term in all texts, parityequality can be used to describe the equivalence of two integers under parity.

Formally, parityequality between integers a and b is defined by a ≡ b modulo 2. Equivalently, a and b have the same remainder when divided by 2. This relation partitions the set of all integers into two parity classes: the even numbers {2k | k ∈ Z} and the odd numbers {2k+1 | k ∈ Z}.

As an equivalence relation, parityequality is reflexive (a ~ a for all a), symmetric (if a ~ b then b ~ a), and transitive (if a ~ b and b ~ c then a ~ c). The two resulting equivalence classes are the even and odd integers, and every integer belongs to exactly one class.

Key implications include that the sum of two parity-equal integers is even, since even + even = even and odd + odd = even. Moreover, if a ~ b, then for any integer c, a + c ~ b + c; adding the same integer to both preserves parityequality. Parityequality is a natural expression of arithmetic modulo 2 and underpins many results in modular arithmetic and parity-based reasoning.

See also: parity (even and odd), congruence modulo 2, equivalence relation, parity bit.