# General Antiperiodicity Property

## Theorem

Let $f: X \to X$ be an antiperiodic function, where $X$ is either $\R$ or $\C$.

Let $L$ be an antiperiodic element of $f$.

Let $n \in \Z$ be an integer.

If $n$ is even, then $n L$ is a periodic element of $f$.
If $n$ is odd, then $n L$ is an antiperiodic element of $f$.

## Proof

Suppose that $X = \C$.

### Case 1

If $n$ is even, then:

 $\ds \map f {x + n L}$ $=$ $\ds \map f {x + \paren {2 k} L}$ for some $k \in \Z$ $\ds$ $=$ $\ds \map f {x + \paren {k 2} L}$ Complex Multiplication is Commutative $\ds$ $=$ $\ds \map f {x + k \paren {2 L} }$ Complex Multiplication is Associative $\ds$ $=$ $\ds \map f x$ by Double of Antiperiodic Element is Periodic and the General Periodicity Property

$\Box$

### Case 2

If $n$ is odd, then:

 $\ds \map f {x + n L}$ $=$ $\ds \map f {x + \paren {2 k + 1} L}$ for some $k \in \Z$ $\ds$ $=$ $\ds \map f {x + \paren {\paren {2 k} L + L} }$ Complex Multiplication Distributes over Addition $\ds$ $=$ $\ds \map f {\paren {x + \paren {2 k} L} + L}$ Complex Addition is Associative $\ds$ $=$ $\ds -\map f {x + \paren {2 k} L}$ $\ds$ $=$ $\ds -\map f x$ by Case 1

The proof for when $X = \R$ is nearly identical.

$\blacksquare$