Is R an Equivalence Relation in R x R Based on a Quadratic Equation?

In summary: Reason: "s" and "t" are not used in the original problem statement, while "a" and "b" are (they represent positive numbers). Using "s" and "t" as temporary variable names is less likely to cause confusion.
  • #1
nikie1o2
7
0
In R x R , ley (x,y) R (u,v) if ax^2 +by^2=au^2 + bv^2, where a,b >0. Determine the relation R is an equivalnce relation. Prove or give a counter example
 
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  • #2
Welcome to PF!

Hi nikie1o2! Welcome to PF! :smile:

(try using the X2 tag just above the Reply box :wink:)

Tell us how far you've got, and where you're stuck, and then we'll know how to help! :smile:
 
  • #3


tiny-tim said:
Hi nikie1o2! Welcome to PF! :smile:

(try using the X2 tag just above the Reply box :wink:)

Tell us how far you've got, and where you're stuck, and then we'll know how to help! :smile:

Hello, thank you for the warm welcome.

I have done equivalence relations before but with just two variables not 4. So i was confused on how to prove the reflexive, symmetric & transitive properties.

For reflexive i was thinking if (x,y)R(x,y) then ax^2+by^2=ax^2+by^2- so that is true
Symmetry: (x,y)R(u,v) then (u,v)R(x,y) is true

For Transitive i knoe if(x,y)R(u,v) and (u,v)R(a,b), then (x,y)R(a,b). I am just confused on how to show the equations for that and that it's true...
 
  • #4
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  • #5
Hello nikie1o2! :smile:

(Please use the X2 tag just above the Reply box :wink:)
nikie1o2 said:
For Transitive i knoe if(x,y)R(u,v) and (u,v)R(a,b), then (x,y)R(a,b). I am just confused on how to show the equations for that and that it's true...

Just write out the definitions of (x,y)R(u,v) and (u,v)R(a,b) … then it should be obvious! :smile:

(btw, the equivalence classes are a well-known geometrical shape … can you se which?)
 
  • #6
Just a comment: using "a" and "b" is a bad choice of variable names in this problem. May I suggest using "s" and "t" instead? I.e., use
(u,v)R(s,t)​
instead of
(u,v)R(a,b)​
when working out the transitive property.
 

Related to Is R an Equivalence Relation in R x R Based on a Quadratic Equation?

What is a proof equivalence relation?

A proof equivalence relation is a mathematical concept that describes a relationship between elements of a set. It states that a set is divided into subsets, and each subset contains elements that are related to each other in a specific way.

How is a proof equivalence relation different from other types of relations?

A proof equivalence relation is different from other types of relations because it has three specific properties: reflexivity, symmetry, and transitivity. These properties ensure that the relationship between elements is well-defined and consistent.

What is the importance of proof equivalence relations in mathematics?

Proof equivalence relations are important in mathematics because they help us classify and understand the relationships between elements in a set. They also allow us to make logical deductions and proofs, which are essential in many mathematical fields.

Can you give an example of a proof equivalence relation?

One example of a proof equivalence relation is the "is congruent to" relation in geometry. If two triangles have equal sides and angles, they are considered equivalent, and this relationship is reflexive, symmetric, and transitive.

Are there any real-life applications of proof equivalence relations?

Yes, there are several real-life applications of proof equivalence relations. They are used in fields such as computer science, data analysis, and social sciences to classify and organize data, make logical deductions, and establish relationships between different entities.

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