# arrow_back What is an abstraction of a concept $C$?

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What is an abstraction of a concept $C$?

An abstraction of a concept $C$ is a concept $C^{\prime}$ that includes all instances of $C$ and that is constructed by taking as axioms certain assertions that are true of all instances of $C . C$ may already be defined mathematically, in which case the abstraction is typically a legitimate generalization of $C$. In other cases, $C$ may be a familiar concept or property that has not been given a mathematical definition.

In that case, the mathematical definition may allow instances of the abstract version of $C$ that were not originally thought of as being part of $C$.

Example 1

The concept of "group" is historically an abstraction of the concept of the set of all symmetries of an object. The group axioms are all true assertions about symmetries when the binary operation is taken to be composition of symmetries.

Example 2

The $\epsilon-\delta$ definition of continuous function is historically an abstraction of the intuitive idea that mathematicians had about functions that there was no "break" in the output. This abstraction became the standard definition of "continuous", but allowed functions to be called continuous that were not contemplated before the definition was introduced.

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