= Cardinality of integer-periodic function spaces
For fixed positive integer period $n$, integer-valued <periodic functions> on $\mathbb Z$ are in <bijection> with $\mathbb Z^n$ and hence form a <countable set>. Their union over periods is a <countable union of countable sets>. In contrast, period-one integer-valued <functions> on $\mathbb Q$ can prescribe arbitrary binary values on its infinitely many distinct classes modulo $\mathbb Z$. The <power set> of a countably infinite set injects into them, proving uncountability.
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