equivalence relation

andlook

Hey

I can't see how x~x+1 is an equivalence relation on the real numbers?

I don't understand what the relation is. Can anyone help?

g_edgar

Perhaps you want the equivalence relation generated by this...

x ~ x+1 ~ x+2 ~ ...

Or perhaps not. Without context, we can only guess.

LumenPlacidum

A relation is a rule that relates elements of one set (in this case the real numbers) to elements of another set (in this case, ALSO the real numbers). An equivalence relation is one from a set to itself that has three properties:

The relation must be reflexive (x must be related to x)
The relation must be symmetric (if x is related to y, then y is related to x), and
The relation must be transitive (if x is related to y, and y is related to z, then x is related to z)

The relation you have, where x is related to x+1 is not an equivalence relation.
1 is not related to 1, since 1 is only related to 2 (not reflexive).
1 is related to 2, but 2 is not related to 1 (it's related to 3) (not symmetric).
1 is related to 2 and 2 is related to 3, but 1 is not related to 3 (not transitive).

It actually fails every single one of the properties.

CRGreathouse

My guess: it's the real numbers mod 1.

andlook

Oops yeah should have been a lot more specific. Context:

Talking about the quotient space of r by the equivalence relation x ~ x+1.

Relating each point to the point +1 ?

1~2 and 2~3 but 1~3 is false...

This is an equivalence relation since states so in literature. So it is clear I don't understand how equivalence relations are working here. Any help? Thanks

LumenPlacidum

Ah, then CRGreathouse was right; the quotient space on that relation is constructing the real numbers modulo 1. When it's flat out saying that that's an equivalence relation, then it's saying that it's reflexive, transitive and symmetric. Essentially, this is saying that the part of any real number before the decimal point doesn't matter, so the equivalence classes (all the things that are equivalent under this relation) are things like Z+{0.5} = {..., -3.5, -2.5, -1.5, -0.5, 0.5, 1.5, 2.5, 3.5, ...}, because all those things are related to each other.

andlook

ok so R / ~ = [0,1)?