Equivalence relation
From Knowino
(Difference between revisions)
| [checked revision] | [pending revision] |
(→Examples: correction) |
(entry) |
||
| Line 6: | Line 6: | ||
* <math>\sim\,</math> is ''transitive'': <math>x \sim y,~y\sim z \Rightarrow x \sim z</math>. | * <math>\sim\,</math> is ''transitive'': <math>x \sim y,~y\sim z \Rightarrow x \sim z</math>. | ||
| − | An '''equivalence class''' for <math>\sim\,</math> is the set of elements of ''X'' all related to some particular element | + | An '''{{Here|equivalence class}}''' for <math>\sim\,</math> is the set of elements of ''X'' all related to some particular element |
:<math>[x]_\sim = \{ y \in X : x \sim y \} . \,</math> | :<math>[x]_\sim = \{ y \in X : x \sim y \} . \,</math> | ||
The equivalence classes form a [[partition]] of the set ''X'', that is, two classes <math>[x]_\sim</math> and <math>[y]_\sim</math> are either equal (have the same members), which is the case when <math>x \sim y\,</math>, or are [[disjoint sets|disjoint]] (have no members in common), which is the case when <math>x \not\sim y</math>. | The equivalence classes form a [[partition]] of the set ''X'', that is, two classes <math>[x]_\sim</math> and <math>[y]_\sim</math> are either equal (have the same members), which is the case when <math>x \sim y\,</math>, or are [[disjoint sets|disjoint]] (have no members in common), which is the case when <math>x \not\sim y</math>. | ||
