Equivalence Relation: Can't Understand x~x+1 on Real Numbers

In summary, the conversation discusses the topic of equivalence relations on the real numbers, specifically the relation x~x+1. It is determined that this relation does not meet the criteria for an equivalence relation. The conversation then delves into the concept of quotient spaces and how they relate to this relation. It is concluded that the quotient space on this relation constructs the real numbers modulo 1, where the part before the decimal point is not considered. The conversation ends with a clarification that R/~ is equivalent to the interval [0,1).
  • #1
andlook
33
0
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?
 
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  • #2
Perhaps you want the equivalence relation generated by this...

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

Or perhaps not. Without context, we can only guess.
 
  • #3
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.
 
  • #4
g_edgar said:
Without context, we can only guess.

My guess: it's the real numbers mod 1.
 
  • #5
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
 
  • #6
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.
 
  • #7
ok so R / ~ = [0,1)?
 

1. What is an equivalence relation?

An equivalence relation is a mathematical concept that describes a relationship between two elements of a set. It is a special type of relation where every element in the set is related to itself, and if two elements are related, then any element related to one of them is also related to the other.

2. What is the meaning of x~y in an equivalence relation?

The symbol "~" in an equivalence relation denotes that the elements x and y are related. This means that they satisfy the properties of reflexivity, symmetry, and transitivity.

3. How do you understand x~x+1 on real numbers?

In this case, x~x+1 means that the two real numbers x and x+1 are related in an equivalence relation. This means that x is related to x+1, and vice versa, because they are one unit apart on the real number line.

4. Why is it important to understand x~x+1 on real numbers?

Equivalence relations are essential in mathematics as they allow us to classify elements of a set into distinct groups. Understanding x~x+1 on real numbers helps us see that any real number can be represented as a group of numbers related to each other by a constant value of 1.

5. How can I apply the concept of equivalence relation to real-world situations?

Equivalence relations can be applied to real-world situations to categorize objects or events that share common characteristics. For example, we can use the equivalence relation "is equal to" to group objects of the same size together, or the relation "is similar to" to group events that have similar outcomes.

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