Definition:Initial Segment of Natural Numbers/One-Based

Definition

Let $n \in \N$ be a natural number.

The initial segment of the non-zero natural numbers determined by $n$:

$\set {1, 2, 3, \ldots, n}$

is denoted $\N^*_{\le n}$.

Also denoted as

The usual notation for this is $\N^*_n$, but the notation $\N^*_{\le n}$ is less ambiguous.

Some sources use the notation of integers, and denote $\set {1, 2, 3, \ldots, n}$ as $\map \Z n$.

Some sources use $P$ or a variant, for example Undergraduate Topology by Robert H. Kasriel, who uses $\mathbf P_n$.

There does not seem to be any notation for this concept which is neither ambiguous nor cumbersome, so on $\mathsf{Pr} \infty \mathsf{fWiki}$ it is commonplace to select a simple notation and define it at point of use.

Also see

• Definition:Initial Segment of Zero-Based Natural Numbers

Sources

• 1965: J.A. Green: Sets and Groups ... (previous) ... (next): $\S 1.1$. Sets: Example $7$
• 1971: Allan Clark: Elements of Abstract Algebra ... (previous) ... (next): Chapter $1$: Mappings: $\S 15$
• 1971: Robert H. Kasriel: Undergraduate Topology ... (previous) ... (next): Notation for Some Important Sets
• 1971: Robert H. Kasriel: Undergraduate Topology ... (previous) ... (next): $\S 1.8$: Collections of Sets: Definition $8.4$