Equivalence Relation on ℝ: xRy if x≥y | Symmetry and Transitivity Explained

Then try to prove it using the property itselfIn summary, The given relation is an equivalence relation on the set, with the partition being x≥y. It is reflexive because x≥x, and it is symmetric and transitive because for any x,y in X, if x≥y, then y≥x and for any x,y,z in X, if x≥y and y≥z, then x≥z. To prove this, try substituting different values for x,y,z and see if the property holds, then use the property itself to prove it.
  • #1
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Homework Statement



Determine whether the given relation is an equivalence relation on the set. Describe the partition arising from each equivalence relation:

xRy in ℝ if x≥y

Homework Equations



Reflexive: for all x in X, x~x
Symmetric: for all x,y in X, if x~y, then y~x
Transitive: for all x,y,z in X, if x~y, and y~z, then x~z

The Attempt at a Solution



I showed that it's reflexive, because x≥x

I'm kind of confused in regards to how to show that it's symmetric and transitive (if it is)?

Any help is appreciated, thanks.
 
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  • #2
The first thing i would do it's to try some value of x y and z to see if the property holds.
 

1. What is an equivalence relation?

An equivalence relation is a mathematical concept that defines a relationship between elements of a set. It is a binary relation that follows three properties: reflexivity, symmetry, and transitivity.

2. How is an equivalence relation different from other types of relations?

An equivalence relation differs from other types of relations, such as partial orders or total orders, in that it does not impose any kind of hierarchy or ranking among the elements of a set. Instead, it focuses on establishing equality or similarity between elements.

3. What is an example of an equivalence relation?

An example of an equivalence relation is the relation of congruence in geometry. Two geometric figures are considered congruent if they have the same size and shape, which satisfies the three properties of an equivalence relation.

4. What are some real-world applications of equivalence relations?

Equivalence relations have various applications in fields such as computer science, linguistics, and social sciences. For example, in computer science, equivalence relations are used to classify data and identify patterns. In linguistics, equivalence relations help in understanding the relationships between different languages. In social sciences, they are used to group individuals based on similar characteristics.

5. How are equivalence relations useful in mathematics?

Equivalence relations are useful in mathematics as they provide a way to classify and group objects based on specific criteria. This allows for easier analysis and understanding of complex mathematical structures. Additionally, equivalence relations are often used as a tool to prove theorems and solve problems in various mathematical fields.

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