Difference between revisions of "Equivalence relation"

Revision as of 00:44, 24 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)

Revision as of 21:20, 23 June 2006 (view source) Mathjoker (talk \| contribs) ← Older edit		Revision as of 00:44, 24 June 2006 (view source) ComplexZeta (talk \| contribs) m Newer edit →
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−	Let <math>S</math> be a set. A relation <math>\sim</math> on <math>S</math> is said to be an ''equivalence relation'' if <math>\sim</math> satisfies the following three properties:	+	Let <math>S</math> be a set. A relation <math>\sim</math> on <math>S</math> is said to be an '''equivalence relation''' if <math>\sim</math> satisfies the following three properties:

	1. For every element <math>x \in S</math>, <math>x \sim x</math>. (Reflexive property)		1. For every element <math>x \in S</math>, <math>x \sim x</math>. (Reflexive property)