least element

USAGE SUMMARY

The phrase "least element" is correct and usable in written English.
It is typically used in mathematical or computer science contexts to refer to the smallest value in a set or collection. Example: "In the set of numbers {3, 1, 4, 2}, the least element is 1."

✓ Grammatically correct

Science

Encyclopedias

News & Media

Human-verified examples from authoritative sources

Exact Expressions

41 human-written examples

An ordered set is said to be well-ordered if every nonempty subset has a least element.

Encyclopedia Britannica

And if they want to talk about whatever is troubling them, there are plenty of well-trained ears at their disposal, but not the least element of compunction.

Independent

Caroline Lucas, leader of the Green Party and the UK's first Green MP, responded yes to all three questions but emphasised the 'at least' element of the 20% cuts by 2020.

The Guardian

The origin of the axiom of choice was Cantor's recognition of the importance of being able to "well-order" arbitrary sets i.e., to define an ordering relation for a given set such that each nonempty subset has a least element.

Encyclopedia Britannica

If there exists a greatest element of one set or a least element of the other set, then the cut defines a rational number; otherwise the cut defines an irrational number.

Encyclopedia Britannica

The number $0$ is the least element of $\mathbb{N}$, but $\mathbb{Z}$ has no least element.

SEP

Human-verified similar examples from authoritative sources

Similar Expressions

19 human-written examples

Also new is the system – or at least elements of it.

The New York Times

At least elements to calculate the cost of the plant, for both reactors are given.

Energy

The signals of us moving to at least elements of this world are already evident, particularly in a UK context.

The Guardian

It could do enough damage that you would need to replace the boiler or at least elements of it.

The New York Times - Magazine

Browsers speed up Web use by storing, or caching, recently viewed Web pages, or at least elements of them, on their user's hard drive.

The New York Times - Tech

Expert writing Tips

Best practice

When using "least element" in mathematical or computer science writing, ensure that the context clearly establishes the ordering relation being used.

Common error

Avoid using "least element" when referring to partially ordered sets where multiple elements might be incomparable. In such cases, "minimal element" is more appropriate.

Linguistic Context

The phrase "least element" functions as a noun phrase, typically acting as a subject or object in mathematical or logical statements. Ludwig indicates that it is commonly used in technical and scientific writing.

Expression frequency: Common

Frequent in

Science

60%

Encyclopedias

20%

News & Media

10%

Less common in

Wiki

5%

Formal & Business

3%

Reference

2%

Ludwig's WRAP-UP

In summary, "least element" is a noun phrase predominantly used in formal and scientific contexts to denote the smallest value within a set. Ludwig AI indicates that the phrase is grammatically correct and commonly used. While alternatives like "minimum element" and "smallest element" exist, "least element" maintains a specific technical connotation, particularly in mathematical and logical discourse. It's crucial to differentiate it from "minimal element" when discussing partially ordered sets. The primary usage contexts, as evidenced by Ludwig, include science and encyclopedias, highlighting its role in defining mathematical structures and relationships.

More alternative expressions

Phrases that express similar concepts, ordered by semantic similarity:

minimum element

This alternative directly replaces "least" with its synonym "minimum", maintaining the exact meaning.

smallest element

This phrase replaces "least" with "smallest", offering a more common and easily understandable alternative.

minimal element

While similar, "minimal element" can sometimes refer to an element that is not greater than any other element in a partially ordered set, differing slightly from "least element" in a totally ordered set.

lowest element

This alternative uses "lowest" instead of "least", maintaining the same meaning, particularly in numerical contexts.

smallest member

This alternative replaces "element" with "member", providing a slightly less formal phrasing.

bottom element

This phrase is often used in the context of lattices and ordered sets to denote the smallest element.

infimum

"Infimum" is a more technical term for the greatest lower bound, which coincides with the least element if it exists within the set.

lower bound

While not a direct synonym, the "least element" is a specific type of lower bound.

initial element

In some contexts, especially when discussing sequences or ordered lists, the "initial element" can refer to the "least element".

first element

Similar to "initial element", "first element" implies an ordering and the selection of the smallest one.

FAQs

What is the difference between "least element" and "minimum element"?

The terms "least element" and "minimum element" are often used interchangeably, especially when referring to totally ordered sets. However, in some contexts, "minimum element" might be used more broadly.

How do you identify the "least element" in a set?

To find the "least element", you need to compare all the elements in the set according to a defined ordering. The element that is smaller than or equal to all other elements is the "least element".

Can a set have more than one "least element"?

No, a set can have at most one "least element". If there were two distinct elements both smaller than or equal to all others, they would have to be equal to each other.

What if a set doesn't have a "least element"?

If a set doesn't have a "least element", it means there isn't an element that is smaller than or equal to all other elements in the set. The set might have a greatest lower bound (infimum), but it's not an element of the set.