Difference between revisions of "Equivalence relation"

Latest revision as of 11:51, 22 May 2007

Let $S$ be a set. A binary relation $\sim$ on $S$ is said to be an equivalence relation if $\sim$ satisfies the following three properties:

1. For every element $x \in S$ , $x \sim x$ . (Reflexive property)

2. If $x, y \in S$ such that $x \sim y$ , then we also have $y \sim x$ . (Symmetric property)

3. If $x, y, z \in S$ such that $x \sim y$ and $y \sim z$ , then we also have $x \sim z$ . (Transitive property)

Some common examples of equivalence relations:

The relation $=$ (equality), on the set of real numbers.
The relation $\cong$ (congruence), on the set of geometric figures in the plane.
The relation $\sim$ (similarity), on the set of geometric figures in the plane.
For a given positive integer $n$ , the relation $\equiv \pmod n$ , on the set of integers. (Congruence mod n)

@@ Line 1: / Line 1: @@
 Let <math>S</math> be a [[set]].  A [[binary relation]] <math>\sim</math> on <math>S</math> is said to be an '''equivalence relation''' if <math>\sim</math> satisfies the following three properties:
-. For every element <math>x \in S</math>, <math>x \sim x</math>.  (Reflexive property)
+. For every element <math>x \in S</math>, <math>x \sim x</math>.  ([[Reflexive property]])
-. If <math>x, y \in S</math> such that <math>x \sim y</math>, then we also have <math>y \sim x</math>.  (Symmetric property)
+. If <math>x, y \in S</math> such that <math>x \sim y</math>, then we also have <math>y \sim x</math>.  ([[Symmetric property]])
-. If <math>x, y, z \in S</math> such that <math>x \sim y</math> and <math>y \sim z</math>, then we also have <math>x \sim z</math>.  (Transitive property)
+. If <math>x, y, z \in S</math> such that <math>x \sim y</math> and <math>y \sim z</math>, then we also have <math>x \sim z</math>.  ([[Transitive property]])