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
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.
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.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) ?
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.
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.
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.
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.
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.