Proving ~ is an Equivalence Relation of A

In summary: So for reflexive, you need to show that (u,v,w) ~ (u,v,w) for any values of u, v, and w. This can be done by choosing k=1. Therefore, the relation is reflexive.
  • #1
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Homework Statement



Consider the set A={(u,v,w) in R^3 : u^2+v^2>0} and define a relation ~ on A by (u,v,w)~(u',v',w') IFF there exists a "k" in R, where k doesn't equal 0: (u',v',w')=(ku,kv,kw)

Prove that ~ is an equivalence relation of A


Homework Equations


I honestly don't know where to start, i know i need to satisfy the reflexive, symmetric & transitive requirements but i don't even know what the relation is here. Any help is very much appreciated.


The Attempt at a Solution

 
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  • #2
You're given the relation. Two vectors (u,v,w) and (u',v',w') are related if they are non-zero multiples of each other. For example, (1,1,1) ~ (2,2,2) since (2,2,2)=(2x1, 2x1, 2x1), i.e. k=2.
nikie1o2 said:

Homework Statement



Consider the set A={(u,v,w) in R^3 : u^2+v^2>0} and define a relation ~ on A by (u,v,w)~(u',v',w') IFF there exists a "k" in R, where k doesn't equal 0: (u',v',w')=(ku,kv,kw)

Prove that ~ is an equivalence relation of A


Homework Equations


I honestly don't know where to start, i know i need to satisfy the reflexive, symmetric & transitive requirements but i don't even know what the relation is here. Any help is very much appreciated.


The Attempt at a Solution

 
  • #3
vela said:
You're given the relation. Two vectors (u,v,w) and (u',v',w') are related if they are non-zero multiples of each other. For example, (1,1,1) ~ (2,2,2) since (2,2,2)=(2x1, 2x1, 2x1), i.e. k=2.

Ok, So to show its reflexive i can pick an real numbers? I can say (u,v,w)=(1,1,1) so (1,1,1) R (1,1,1) ?
 
  • #4
No, for reflexive, you'd need to look at a general triple (u,v,w). What does it mean to have
(u,v,w) ~ (u,v,w) ?

Write it out and find a value for k.

You'll need to work with general triples for all of the axioms for an equivalence relation.
(if you really get stuck, you might work with specific points to get a handle on what's happening, but you can't choose specific points for the proof)
 
  • #5
nikie1o2 said:
Ok, So to show its reflexive i can pick an real numbers? I can say (u,v,w)=(1,1,1) so (1,1,1) R (1,1,1) ?
No, I was just using specific numbers as an example. As Mathnerdmo said, for a proof, you need to show it's true for the general case.
 

1. What is an equivalence relation?

An equivalence relation is a mathematical concept that defines a relation between elements of a set. It is reflexive, symmetric, and transitive, meaning that it satisfies the properties of reflexivity, symmetry, and transitivity.

2. How do you prove that a relation is an equivalence relation?

To prove that a relation is an equivalence relation, you must show that it satisfies the three properties of reflexivity, symmetry, and transitivity. This can be done by using mathematical proofs or counterexamples.

3. What is the importance of proving an equivalence relation?

Proving an equivalence relation is important because it helps us to understand the relationships between elements in a set. It also allows us to partition a set into equivalence classes, which can be useful in various fields of mathematics, such as algebra and topology.

4. Can an equivalence relation be defined on any set?

Yes, an equivalence relation can be defined on any set, as long as the three properties of reflexivity, symmetry, and transitivity are satisfied. However, not all relations on a set are equivalence relations.

5. What are some examples of equivalence relations?

Some examples of equivalence relations include the relation "is equal to" on the set of integers, the relation "is congruent modulo n" on the set of integers, and the relation "is parallel to" on the set of lines in a plane.

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