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 Math Help > Number Theory > Definitions in Number Theory > Equivalence Relation

### Equivalence Relation

A student writes,

Some of the stuff I'm working on right now involves 'equivalence relations'. I have seen that they must be transitive, symmetrical and reflexive, but I don't understand any of these words, or the importance of an equivalence relation, or what an equivalence relation really is at all!
Could I have an explanation please?

First, a little terminology. If we have a relation, R, and elements a and b of a set, then we can say aRb to mean that the relation holds (i.e. is true) of elements a and b, in that order.

Transitive means: if aRb and bRc then aRc.
Example: the relation > (greater than) is transitive. if a>b and b>c then a>c.

Symmetric means: if aRb then bRa.
Example: the relation != (not equal to) is symmetric. if a!=b then b!=a.

Reflexive means: aRa
Example: the relation >= (greater than or equal) is reflexive. a>=a.

None of these examples, however, is an equivalence relation, because in each case I chose an example in which the particular property applies, but not all three properties.

### Related pages in this website

Set Theory - Groups, group operation has properties: Closure, Associativity, Identity, Inverse.

The webmaster and author of this Math Help site is Graeme McRae.