# 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 $n$:

- $\set {0, 1, 2, \ldots, n - 1}$

is denoted $\N_{< n}$.

### One-Based

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 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:

- $\map \Z n = \set {1, 2, \ldots, n} = \set {z \in \Z: 1 \le z \le n}$

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 $S_n = \set {1, 2, \ldots, n - 1}$.

## Sources

- 1971: Robert H. Kasriel:
*Undergraduate Topology*... (previous) ... (next): $\S 1.8$: Collections of Sets: Definition $8.4$ - 2000: James R. Munkres:
*Topology*(2nd ed.) ... (previous) ... (next): $1$: Set Theory and Logic: $\S 4$: The Integers and the Real Numbers