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\({\mathbb{N}}\) is the set of natural numbers. We will use the convention in which natural numbers start from 0, so \({\mathbb{N}}= \{0, 1, 2 \ldots \}\).
\({\mathbb{Q}}\) is the field of rational numbers, \({\mathbb{R}}\) is the field of real numbers.
For \(F \in \{{\mathbb{Q}},{\mathbb{R}}\}\), \(F^{>0} := \{x \in F \mid x > 0\}\), \(F^{\geq 0} := \{x \in F \mid x \geq 0\}\).
\(\log: {\mathbb{R}}^{\geq 0} \rightarrow {\mathbb{R}}\sqcup \{-\infty\}\) will denote the logarithm in base 2.