Interpreting equivalence classes geometrically

In summary, the conversation discusses defining a relation ~ for (x,y) and (u,v) in R2, where (x,y) is equivalent to (u,v) if x2+y2 = u2+v2. The conversation then goes on to prove that ~ defines an equivalence relation on R2 and interpret the equivalence classes geometrically as points on a circle about the origin. The conversation also discusses the reflexivity, symmetry, and transitivity of the relation and how it relates to triangle congruence and similarity.
  • #1
MathDude
8
0

Homework Statement



For (x, y) and U, v) in R2, define (x,y)~(u,v) if x2+y2 = u2+v2. Prove that ~ defines an equivalence relation on R2 and interpret the equivalence classes geometrically.

Homework Equations



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The Attempt at a Solution



The first part is easy. I proved transitivity, reflexivity, and symmetry as per the definition of an equivalence. I'm a little confused how to do so geometrically. Since x2 + y2 can represent the Pythagorean identity, I'm assuming the proof involves triangles. I understand that the properties of an equivalency (trans, reflect, and sym) can resemble triangle congruence and similarity, but I'm not quite so sure how to apply it. Any hints would be helpful.
 
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  • #2
consider the distance a point (x,y) is from the origin...
 
  • #3
Look at the set of all (x,y) so that x2+y2=1. Is this an equivalence class? What else is it?
 
  • #4
That makes sense conceptually, but I'm not quite sure how to apply it. If we're letting x2+y2 equal to diameter one, what happens to our (u, v)? I'm confused how to get this to work. What do we define the relation as?
 
  • #5
say you have any (x,y) with x^2 + y^2 = 1

and any (u,v) with u^2 + v^2 = 1

then based on your definition
(x,y) ~ (u,v)

now on what geomertic object do they both lie?
 
  • #6
They both lie on a circle. I can prove they're equal through distance formula at any given point of origin.

How do you geometrically interpret reflexivity, symmetry, and transitivity? I mean, it's obvious to me visually, but how do you prove it?



Story of my life: "It's obvious conceptually, but how do I prove it?"
 
  • #7
Could you just draw a (x,y) and (u,v) circles and show their radii and origins are identical and say its obviously reflexive, symmetric, and transitive?
 
  • #8
sorry misread your question

it doesn't ask you to prove geometrically the equivalence realtion, just intepret what it is, which you have already done

each equivalence class, label it by r>0, corresponds to all points at a distance r form the origin, which as a set is the points contained in the circle of radius r, around the origin
 
  • #9
MathDude said:
Could you just draw a (x,y) and (u,v) circles and show their radii and origins are identical and say its obviously reflexive, symmetric, and transitive?
(x,y) and (u,v) are not circles- they are points on a circle. Geometrically, two points, (x,y) and (u,v) are equivalent if and only if they lie on the same circle about the origin.
 

FAQ: Interpreting equivalence classes geometrically

1. What are equivalence classes in geometry?

Equivalence classes in geometry are sets of points or figures that have the same properties or characteristics. These classes are often used to group geometric objects that are considered equivalent or indistinguishable from each other.

2. How are equivalence classes determined?

Equivalence classes are determined by defining an equivalence relation, which specifies the conditions under which two objects can be considered equivalent. In geometry, this can be based on criteria such as shape, size, orientation, or other geometric properties.

3. What is the significance of interpreting equivalence classes geometrically?

Interpreting equivalence classes geometrically allows us to better understand the relationships between different geometric objects. By grouping them into equivalence classes, we can identify patterns and make generalizations about the properties and behavior of these objects.

4. Can equivalence classes be applied to all types of geometry?

Yes, equivalence classes can be applied to various types of geometry, including Euclidean, non-Euclidean, and projective geometry. However, the criteria for determining equivalence may differ depending on the specific type of geometry.

5. What are some real-world applications of equivalence classes in geometry?

Equivalence classes in geometry have various practical applications, such as in computer graphics, pattern recognition, and data analysis. They are also used in fields such as architecture, engineering, and physics to understand and model geometric structures and systems.

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