What's the difference between an equality and an equivalence

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SUMMARY

The discussion clarifies the distinction between equality and equivalence in mathematics, emphasizing that equality is a specific case of equivalence. An equivalence relation is defined on a set, where two elements are considered equal with respect to that relation. For instance, in elementary algebra involving real numbers, the equivalence relation aligns with standard arithmetic, allowing the use of the equality symbol "=" without further specification. This understanding is crucial for mathematical precision and context.

PREREQUISITES
  • Understanding of basic mathematical concepts, including sets and relations.
  • Familiarity with the definition of equivalence relations in mathematics.
  • Knowledge of elementary algebra and real numbers.
  • Ability to interpret mathematical notation and symbols.
NEXT STEPS
  • Study the formal definition of equivalence relations in set theory.
  • Explore examples of different equivalence relations on various sets.
  • Learn about the properties of equivalence relations, such as reflexivity, symmetry, and transitivity.
  • Investigate the implications of equivalence classes in mathematical contexts.
USEFUL FOR

Students of mathematics, educators teaching algebra and set theory, and anyone interested in the foundational concepts of mathematical relations.

PsychonautQQ
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Might sound like a stupid question, but if somebody could give me a mathematical description of the difference between equality and equivalence that might be really interesting.
 
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There is a standard mathematical definition for an "equivalence relation". Are you familiar with it?

There can be several different equivalence relations defined on the same set of things. Two things related by a certain equivalence relation R are said to be "equal with respect to R". In a given context it may be clear what equivalence relation is being used. For example, in a textbook discussing elementary algebra with the real numbers, the equivalence relation is understood to be the one we learn in elementary arithmetic. So the book won't bother to say two numbers are "equal with respect to the equivalence relation on the real numbers". It will just say the numbers are "equal" or use the abbreviation "=".
 

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