### 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.

### Internet references

### Related pages in this website

Sets, Set
Description Notation

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

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Graeme McRae.