equivalence relation

USAGE SUMMARY

The phrase "equivalence relation" is correct and usable in written English.
It is typically used in mathematics and computer science to describe a specific type of relation that satisfies reflexivity, symmetry, and transitivity. Example: "In set theory, an equivalence relation groups elements into equivalence classes based on a defined relation."

✓ Grammatically correct

Science

Encyclopedias

Human-verified examples from authoritative sources

Exact Expressions

60 human-written examples

Congruence of triangles is an equivalence relation in geometry.

Encyclopedia Britannica

Identity is an equivalence relation; i.e., it is reflexive, symmetrical, and transitive.

Encyclopedia Britannica

A relation that is reflexive, symmetrical, and transitive is called an equivalence relation.

Encyclopedia Britannica

Equivalence relation, In mathematics, a generalization of the idea of equality between elements of a set.

Encyclopedia Britannica

The equivalence relation he intends is unclear.

SEP

For suppose E is some equivalence relation.

SEP

Classical identity is an absolute equivalence relation.

SEP

This is an equivalence relation between wffs.

SEP

It based on equivalence relation.

Mathematical Sciences

Clearly, ∼ is an equivalence relation.

Journal of Inequalities and Applications

We introduce an equivalence relation between cycles.

Pacific Journal of Mathematics for Industry

Expert writing Tips

Best practice

When using "equivalence relation", ensure you explicitly define the relation and demonstrate that it satisfies the reflexive, symmetric, and transitive properties to avoid ambiguity.

Common error

A common mistake is assuming a relation is an "equivalence relation" without formally verifying that it meets all three requirements: reflexivity, symmetry, and transitivity. Always check each property to maintain mathematical rigor.

Linguistic Context

The phrase "equivalence relation" functions as a noun phrase identifying a specific type of binary relation in mathematics and logic. As Ludwig AI confirms, the phrase is grammatically sound and widely used. Ludwig provides numerous examples demonstrating its use in academic and scientific contexts.

Expression frequency: Very common

Frequent in

Science

72%

Encyclopedias

21%

Formal & Business

2%

Less common in

News & Media

0%

Wiki

0%

Reference

0%

Ludwig's WRAP-UP

The phrase "equivalence relation" is a well-defined mathematical term used to describe a relation that is reflexive, symmetric, and transitive. As Ludwig confirms, its usage is grammatically correct and common, particularly in scientific and encyclopedic contexts.

When using this phrase, ensure you verify that the relation meets all three properties to avoid errors. Related phrases such as "equality relation" or "congruence relation" can be used to express similar concepts with slightly different nuances. Understanding the specific mathematical properties of the described relation will allow you to use the proper wording and avoid any confusion or misinterpretation.

The phrase is most appropriate in formal and scientific writing, maintaining precision and rigor.

More alternative expressions

Phrases that express similar concepts, ordered by semantic similarity:

equality relation

Focuses on the aspect of equality as the core characteristic.

identity relation

Emphasizes the aspect of identity rather than general equivalence.

congruence relation

Highlights the idea of congruence, typically used in geometry or number theory.

similarity relation

Suggests a looser form of equivalence based on shared characteristics.

proportionality relation

Implies a relationship of proportionality between elements.

correspondence relation

Highlights a mutual relationship or connection between items.

isomorphism relation

Refers to a structural equivalence between mathematical objects.

homomorphism relation

Indicates a structure-preserving map between algebraic structures.

relationship of indistinguishability

Emphasizes the lack of perceptible difference between elements under certain conditions.

classification relation

Highlights the action of classifying elements into equivalent groups.

FAQs

How do you define an "equivalence relation"?

An "equivalence relation" is a binary relation that is reflexive, symmetric, and transitive. Reflexive means that every element is related to itself. Symmetric means that if a is related to b, then b is related to a. Transitive means that if a is related to b and b is related to c, then a is related to c.

What are some examples of "equivalence relations"?

Examples of "equivalence relations" include equality (=) on any set, congruence modulo n on integers, and similarity of geometric shapes. These relations satisfy the reflexive, symmetric, and transitive properties.

How does an "equivalence relation" differ from other types of relations?

Unlike other relations, an "equivalence relation" must satisfy all three properties: reflexivity, symmetry, and transitivity. Relations like orderings (e.g., ≤) are transitive and reflexive but not symmetric, while other relations may lack reflexivity or transitivity.

What is the significance of using an "equivalence relation" in mathematics?

An "equivalence relation" allows for partitioning a set into equivalence classes, where elements within the same class share a specific characteristic. This is useful for simplifying complex structures and defining new mathematical objects based on these classes. For example, the concept of "congruence relation" in number theory.