Difference between revisions of "Equivalence relation"

Revision as of 21:20, 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. (Congruence mod $n$ )

@@ Line 12: / Line 12: @@
 * The relation <math>\cong</math> (congruence), on the set of geometric figures in the plane.
 * The relation <math>\sim</math> (similarity), on the set of geometric figures in the plane.
-* For a given positive integer <math>n</math>, the relation <math>\equiv</math> (mod <math>n</math>), on the set of integers.  (Congruence mod <math>n</math>)
+* For a given positive integer <math>n</math>, the relation <math>\equiv</math> (mod <math>n</math>), on the set of integers.  ([[Modular arithmetic|Congruence mod <math>n</math>]])