Class 12 Statistics Notes · GSEB

Neighbourhood

Neighbourhood — learn the δ-neighbourhood of a point in interval, modulus and set forms, plus the punctured neighbourhood that removes the centre. GSEB Class 12 Commerce Statistics notes with interactive number lines.

Last updated: 23 Sep 2026

Notes

What is a Neighbourhood?

δ-neighbourhood of a

The set of all points that lie within a distance δ (delta) of the point a on the real line. It is the open interval (a − δ, a + δ), where δ is a small positive number.

Neighbourhood — Three Forms

N_{\delta}(a) = (a - \delta,\; a + \delta) = \{x : |x - a| < \delta\}

Interval form

(a − δ, a + δ)

From a-step back to a-step forward, endpoints not included.

Modulus form

|x − a| < δ

Distance from x to a is less than δ — the most compact way to write it.

Set form

{x : a − δ < x < a + δ}

Formal set-builder notation — what exam papers often expect.

Google Maps saying you are “within 10 m” of a spot is a δ-neighbourhood of radius 10. Smaller δ = tighter, more precise.

Try It: Neighbourhood Lab

Drag the sliders to move the centre and stretch the radius — all three forms update live.

Neighbourhood Lab — build your own δ-neighbourhood

Centre a = 5

Radius δ = 2

Interval form

(3, 7)

Modulus form

|x − 5| < 2

Set form

{x : |x−5| < 2}

Centre a is included — every point within distance δ of a belongs. All three forms describe the same set.

Solved Examples

Solved Example

Problem

Find the neighbourhood of 4 with radius 0.5, in all three forms.

Solution

Interval (3.5, 4.5) · Modulus |x − 4| < 0.5 · Set {x : 3.5 < x < 4.5}

Solved Example

Problem

If |x − 7| {'<'} 0.1, write the neighbourhood of 7 with radius 0.1 as an interval.

Solution

(6.9, 7.1)

Key Takeaways

A δ-neighbourhood of a = every point within distance δ of a. Three forms, one set: (a − δ, a + δ) = |x − a| < δ = {x : a − δ < x < a + δ}.
δ is a small positive number — it decides how wide the strip around a is (GPS "within 10 m" is δ = 10).
To convert: expand centre ± radius for the interval; wrap |x − a| < δ for the modulus form.
The centre a is always included — the neighbourhood is an open interval around it, nothing removed.