# What does the tilde represent in an equivalence relation on R?

• Robb
In summary, the tilde (~) is used to represent an equivalence relation. It is used to show that two elements, x and y, are related. In part (a) of the attached, it is used to show that x is related to y. The equation ##x \sim y ## should be read as "x is related to y". The purpose of using the tilde is to demonstrate reflexivity, where x~x or x-x is in ##\mathbb Q ##.
Robb
Homework Statement
Prove that tilde is an equivalence relation on R
Relevant Equations
See attached
Can someone please explain what the tilde represents? We have had no info on this to date. I know it has to do with an equivalence relation but not sure what it represents on its own as in part (a) of the attached. Just want to make sure I'm clear. Thanks!

#### Attachments

• hw9.pdf
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Please post the equations here instead of expecting us to download and open some pdf file.

Read ##x \sim y ## as "x is related to y".

To get you started, you need to show (reflexivity) that x~x or x-x is in ##\mathbb Q ## Can you take it from here?

## 1. What is an equivalence relation on R?

An equivalence relation on R is a mathematical concept that defines a relationship between two elements in a set. In this case, the set is the set of real numbers, R. An equivalence relation is a binary relation that satisfies three properties: reflexivity, symmetry, and transitivity.

## 2. How is an equivalence relation different from an equality relation?

An equivalence relation is a broader concept than an equality relation. While an equality relation only considers the exact equality of two elements, an equivalence relation allows for a more general comparison between elements. For example, in an equivalence relation, two elements may be considered equivalent if they are related by a certain property or characteristic, even if they are not exactly equal.

## 3. What are some examples of equivalence relations on R?

Some examples of equivalence relations on R include "is congruent to" for geometric figures, "has the same absolute value as," and "is a multiple of." These relations satisfy the three properties of reflexivity, symmetry, and transitivity.

## 4. How are equivalence relations useful in mathematics?

Equivalence relations are useful in mathematics because they allow us to classify elements in a set into distinct groups based on their relationships with one another. This can help simplify complex problems and make them more manageable. Equivalence relations are also used in various fields of mathematics, such as algebra, topology, and number theory.

## 5. Can an equivalence relation on R be non-binary?

No, an equivalence relation on R must be binary, meaning it relates two elements at a time. This is because the definition of an equivalence relation requires it to satisfy the three properties of reflexivity, symmetry, and transitivity, which can only be applied to two elements at a time.

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