Definition:Equivalence class

From CS2800 wiki
Definition: Equivalence class
If [math]R [/math] is an equivalence relation on a set [math]A [/math], then the equivalence class of [math]a [/math] by [math]R [/math] (written [math]\href{/cs2800/wiki/index.php/%E2%9F%A6a%E2%9F%A7_R}{⟦a⟧_R} [/math]) is the set of elements [math]b \href{/cs2800/wiki/index.php/%E2%88%88}{∈} A [/math] with [math]\href{/cs2800/wiki/index.php/ARb}{aRb} [/math].


In other words, [math]\href{/cs2800/wiki/index.php/%E2%9F%A6a%E2%9F%A7_R}{⟦a⟧_R} \href{/cs2800/wiki/index.php/Definition}{:=} \{b \href{/cs2800/wiki/index.php/%E2%88%88}{∈} A \href{/cs2800/wiki/index.php/%5Cmid}{\mid} \href{/cs2800/wiki/index.php/ARb}{aRb}\} [/math]

Note: I usually use the symbols [ and ] instead of ⟦ and ⟧, but the wiki syntax makes this difficult. You may use either notation.

When [math]R [/math] is clear from context, we just write ⟦a⟧.