Equivalence class

Congruence is an example of an equivalence relation. The leftmost two triangles are congruent, while the third and fourth triangles are not congruent to any other triangle shown here. Thus, the first two triangles are in the same equivalence class, while the third and fourth triangles are each in their own equivalence class.

In mathematics, when a set has an equivalence relation defined on its elements, there is a natural grouping of elements that are related to one another, forming what are called equivalence classes. Notationally, given a set $X$ and an equivalence relation $~$ on $X$ , the equivalence class of an element $a$ in $X$ is the subset of all elements in $X$ which are equivalent to $a$ . It follows from the definition of the equivalence relations that the equivalence classes form a partition of $X$ . The set of equivalence classes is sometimes called the quotient set or the quotient space of $X$ by $~$ and is denoted by $X / ~$ .

单词	Quotient topological space
释义	Quotient topological space 英语百科 Equivalence class In mathematics, when a set has an equivalence relation defined on its elements, there is a natural grouping of elements that are related to one another, forming what are called equivalence classes. Notationally, given a set $X$ and an equivalence relation $~$ on $X$ , the equivalence class of an element $a$ in $X$ is the subset of all elements in $X$ which are equivalent to $a$ . It follows from the definition of the equivalence relations that the equivalence classes form a partition of $X$ . The set of equivalence classes is sometimes called the quotient set or the quotient space of $X$ by $~$ and is denoted by $X / ~$ .
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Quotient topological space

Equivalence class