sets defined as

USAGE SUMMARY

The phrase "sets defined as" is correct and usable in written English.
It can be used when explaining or describing the characteristics or criteria of specific sets in a mathematical or logical context. Example: "In this study, we will analyze the sets defined as all even numbers and all prime numbers."

✓ Grammatically correct

Science

Encyclopedias

Human-verified examples from authoritative sources

Exact Expressions

15 human-written examples

Let U and V be the sets defined as in the proof of Theorem 2.5.

Fixed Point Theory and Applications

Claim 2 Let C 1 + and C 1 + be the sets defined as above.

Advances in Difference Equations

Claim 1 Let S 1 and S 2 be the sets defined as above.

Advances in Difference Equations

Let (mathcal {S}_{1}) and (mathcal{S}_{2}) be the sets defined as above.

Advances in Difference Equations

The set of 2D peaks P can therefore be divided into three disjoint sets defined as follows.

Algorithms for Molecular Biology

Let (mathcal {S}), (mathcal {B}), (partial mathcal {S}), and (partial mathcal {B}) be the sets defined as (1.12) and (1.13).

Boundary Value Problems

Human-verified similar examples from authoritative sources

Similar Expressions

45 human-written examples

For two nonempty disjoint sets define as Let and be the -fields generated by and respectively.

Journal of Inequalities and Applications

In particular, p is valid if the empty set, defined as having no members, logically entails p for this is just another way of saying that p is true in every model.

Encyclopedia Britannica

Let X be a finite set defined as (X={1,2,3,4}).

Fixed Point Theory and Applications

Probe-sets defined as "absent" calls across all the samples were removed from data analysis to reduce the false positives.

BMC Cancer

QLQ-C30 set: defined as all ITT patients having an assessable QLQ-C30 (<50% of missing responses) at baseline, FACIT-CCSQ set: defined as all ITT patients having an assessable FACIT-CCSQ (<50% of missing responses) at baseline.

British Journal of Cancer

Expert writing Tips

Best practice

When using the phrase "sets defined as", ensure that the definition is clear, concise, and unambiguous. This helps avoid confusion and ensures that the reader understands exactly which elements belong to the set.

Common error

Avoid defining sets using overly broad or vague criteria. Always provide precise conditions for set membership to prevent misinterpretations.

Linguistic Context

The phrase "sets defined as" serves as a definitional marker, introducing the criteria or characteristics that specify the members of a particular set. According to Ludwig, this phrase is used correctly and effectively in various contexts. The examples show its common use in academic and scientific writing.

Expression frequency: Common

Frequent in

Science

85%

Encyclopedias

9%

Formal & Business

3%

Less common in

News & Media

1%

Wiki

1%

Social Media

0%

Ludwig's WRAP-UP

In summary, "sets defined as" is a useful phrase for introducing precise definitions of categories, primarily in formal and scientific writing. Ludwig AI indicates the phrase is used correctly and is commonly found in academic and scientific literature. When employing this phrase, ensure that the definition is unambiguous and clear to avoid confusion. Alternative phrasing includes "sets characterized as" or "sets specified as", but "sets defined as" remains a direct and effective choice for clarity.

More alternative expressions

Phrases that express similar concepts, ordered by semantic similarity:

sets characterized as

Focuses on the attributes that distinguish the sets.

sets specified as

Emphasizes the detailed conditions for inclusion in the sets.

sets identified as

Highlights the recognition or labeling of the sets based on criteria.

sets designated as

Suggests a formal assignment or naming of the sets.

sets known as

Indicates common or established names for the sets.

sets classified as

Highlights the grouping or categorization of sets based on shared traits.

sets described as

Focuses on providing a narrative explanation of the sets' attributes.

sets determined by

Emphasizes the conditions influencing set composition.

sets constructed from

Focuses on the components used to create the sets.

sets comprising

Highlights the elements making up the sets.

FAQs

How do I use "sets defined as" in a sentence?

Use "sets defined as" to introduce a clear and concise explanation of what constitutes a particular set. For instance, "The "prime numbers" are the sets defined as all positive integers divisible only by 1 and themselves."

What are some alternatives to "sets defined as"?

Alternatives include phrases like "sets characterized as", "sets specified as", or "sets identified as", depending on the specific nuance you wish to convey.

Is it better to say "sets that are defined as" or "sets defined as"?

"Sets defined as" is more concise and grammatically sound. The inclusion of "that are" is generally unnecessary and can make the sentence sound more verbose. "The "integers" are sets defined as whole numbers, both positive and negative, including zero" is preferred.

Can "sets defined as" be used in non-mathematical contexts?

While "sets defined as" is common in mathematical and scientific contexts, it can be used in other fields where clear categorization is necessary. For example, "The "behavioral traits" are sets defined as observable patterns of action and response."