mathematics

Positively separated sets

Positively separated sets is a mathematics topic covered in the lgStudy science library. This page brings together a partial reference excerpt, illustrations, worked examples, real-world applications and a short study plan, so you can understand Positively separated sets rather than just read about it. In short: In mathematics, two non-empty subsets A and B of a given metric space (X, d) are said to be positively separated if the infimum inf a ∈ A , b ∈ B d ( a , b ) > 0. {\displaystyle \inf _{a\in A,b\in B}d(a,b)>0.} (Some authors also specify that A and B should be disjoint sets; however, this adds nothing to the definition, since if A and B have some common point p, then d(p, p) = 0, and so the infimum above is clearly 0…

Key takeaways

Positively separated sets belongs to mathematics; place it in that map before memorising details.
Learn the definition first, then one example that makes the definition concrete.
Connect Positively separated sets to a quantity you can measure, compute or draw — that is where exam questions come from.
Reproduce the core statement of Positively separated sets from memory before moving on to harder problems.

Reference excerpt

In mathematics, two non-empty subsets A and B of a given metric space (X, d) are said to be positively separated if the infimum

inf a ∈ A , b ∈ B d ( a , b ) > 0. {\displaystyle \inf _{a\in A,b\in B}d(a,b)>0.}

(Some authors also specify that A and B should be disjoint sets; however, this adds nothing to the definition, since if A and B have some common point p, then d(p, p) = 0, and so the infimum above is clearly 0 in that case.) For example, on the real line with the usual distance, the open intervals (0, 2) and (3, 4) are positively separated, while (3, 4) and (4, 5) are not. In two dimensions, the graph of y = 1/x for x > 0 and the x-axis are not positively separated.

References Rogers, C. A. (1998). Hausdorff measures. Cambridge Mathematical Library (Third ed.). Cambridge: Cambridge University Press. pp. xxx+195. ISBN 0-521-62491-6.

Worked examples

Example 1 — a first encounter with Positively separated sets

Start with the simplest possible case. Write down what Positively separated sets claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In mathematics, the smallest case is usually a single object, a single equation or a single measurement. Check that every symbol or term in your sentence has a meaning in that case.

Example 2 — changing one variable

Take the situation from Example 1 and change exactly one quantity: double it, halve it, or set it to zero. Predict what should happen to Positively separated sets before you calculate. Comparing your prediction with the result is the fastest way to find out whether you understand the idea or only the words.

Example 3 — an exam-style question

Typical questions about Positively separated sets ask you to (a) state it precisely, (b) apply it to given data, and (c) explain a limitation. Practise writing all three answers in under five minutes; the third part is what separates a full-mark answer from an average one.

Applications of Positively separated sets

In research: Positively separated sets appears in mathematics research whenever the underlying quantities have to be modelled precisely. Papers usually cite it as a starting assumption and then explore where it breaks down.
In technology and industry: Engineering practice reuses Positively separated sets in design rules, simulations and safety margins. Knowing the idea lets you read a specification sheet and understand why the numbers look the way they do.
In the classroom: Positively separated sets is common in secondary-school and first-year university syllabi. It links to neighbouring topics Metric geometry, Metric geometry stubs, so understanding it makes those chapters shorter.
In everyday life: Look for Positively separated sets outside the textbook — in sport, cooking, traffic, electronics or the sky above you. An example you found yourself is remembered far longer than one you were given.

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How to study Positively separated sets in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Positively separated sets means in your own words.
Compare your version with the excerpt and mark what you missed.
Work through the three examples above with pen and paper.
Explain Positively separated sets out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Positively separated sets in simple terms?

In mathematics, two non-empty subsets A and B of a given metric space (X, d) are said to be positively separated if the infimum inf a ∈ A , b ∈ B d ( a , b ) > 0. {\displaystyle \inf _{a\in A,b\in B}d(a,b)>0.} (Some authors also specify that A and B should be disjoint sets; however, this adds nothi…

Why does Positively separated sets matter?

Because it connects several mathematics ideas at once: it gives you a definition you can apply, a quantity you can calculate, and a way to check whether a result is plausible.

How should I study Positively separated sets?

Read the excerpt, restate it from memory, then work through the examples and applications listed on this page. The five-step study plan above takes about twenty minutes.

What does this page cover?

It gives you a compact reference excerpt plus original lgStudy explanations, examples, applications and study material on Positively separated sets.