Paritypreserving

Paritypreserving is an adjective used across mathematics, computer science, and physics to describe a property of a function, operation, or system that maintains the parity of a quantity, where parity refers to evenness or oddness. In number theory, a function f: Z → Z is paritypreserving if for every integer n, f(n) ≡ n (mod 2). Equivalently, even inputs map to even outputs and odd inputs map to odd outputs.

Examples include the identity function f(n) = n, negation f(n) = -n, and many polynomials with integer coefficients that preserve parity for all integers. In particular, a linear function f(n) = an + b preserves parity exactly when a is odd and b is even, so f(n) ≡ n (mod 2) for all n. More generally, any function constructed so that f(n) ≡ n (mod 2) for all n is paritypreserving.

In discrete computing, parity-preserving operations are those that do not change the parity of a state. In boolean logic and reversible computing, parity-preserving gates map input configurations to outputs with the same parity of 1s, and parity-preserving cellular automata conserve the parity of live cells over time.

In physics and information theory, the concept describes systems with parity symmetry or with parity as a conserved quantity. Parity-preserving transformations are often used in error-detecting codes, cryptographic constructions, and algorithmic design where parity is a useful invariant.