Difference between revisions of "Equivalence relation"
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* The relation <math>\cong</math> (congruence), on the set of geometric figures in the plane. | * 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. | * 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. | + | * 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>) |
Revision as of 22:18, 23 June 2006
Let be a set. A relation
on
is said to be an equivalence relation if
satisfies the following three properties:
1. For every element ,
. (Reflexive property)
2. If such that
, then we also have
. (Symmetric property)
3. If such that
and
, then we also have
. (Transitive property)
Some common examples of equivalence relations:
- The relation
(equality), on the set of real numbers.
- The relation
(congruence), on the set of geometric figures in the plane.
- The relation
(similarity), on the set of geometric figures in the plane.
- For a given positive integer
, the relation
(mod
), on the set of integers. (Congruence mod
)