Equivalence classes in the context of Equivalence relation

Equivalence classes in the context of Equivalence relation

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⭐ Core Definition: Equivalence classes

In mathematics, when the elements of some set $S$ have a notion of equivalence (formalized as an equivalence relation), then one may naturally split the set $S$ into equivalence classes. These equivalence classes are constructed so that elements $a$ and $b$ belong to the same equivalence class if, and only if, they are equivalent.

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⭐ Core Definition: Equivalence classes
Equivalence classes in the context of Up to

Equivalence classes in the context of Up to

Two mathematical objects $a$ and $b$ are called "equal up to an equivalence relation $R$ "

if $a$ and $b$ are related by $R$ , that is,
if $aRb$ holds, that is,
if the equivalence classes of $a$ and $b$ with respect to $R$ are equal.

This figure of speech is mostly used in connection with expressions derived from equality, such as uniqueness or count.For example, " $x$ is unique up to $R$ " means that all objects $x$ under consideration are in the same equivalence class with respect to the relation $R$ .

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