Difference between revisions of "Equivalence relation"

Revision as of 22:18, 23 June 2006

Let $S$ be a set. A 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 $\equiv$ (mod $n$ ), on the set of integers.

@@ Line 2: / Line 2: @@
 . 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, 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)