Definition:Initial Segment of Natural Numbers

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Definition

Zero-Based

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

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

$\left\{{0, 1, 2, \ldots, n-1}\right\}$

is denoted $\N_{< n}$.


One-Based

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

$\left\{ {1, 2, 3, \ldots, n}\right\}$

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


Also denoted as

The usual notation for these are $\N_n$ and $\N^*_n$, but the notations $\N_{< n}$ and $\N^*_{\le n}$ are less ambiguous.


Also defined as

Some sources consider $n$ as an integer and use the symbology:

$\Z \left({n}\right) = \left\{{1, 2, \ldots, n}\right\} = \left\{{z \in \Z: 1 \le z \le n}\right\}$

but this is rare.

Some sources use $\mathbf P_n$ or similar, for either $\N_{< n}$ or $\N^*_{\le n}$, where $\mathbf P$ may stand for positive.

There is considerable inconsistency in the literature. James R. Munkres: Topology (2nd ed.), for example, has $S_n = \left\{{1, 2, \ldots, n - 1}\right\}$.


Sources