Need help please ( Equivalence Relation )

[Mamaxouchk2]

Hello,
Can someone help me with this problem?

I have a set :
A={1,1/3,1/27,1/4,3,1/36,2,2/9,9/4,5}

The Relation is :

∀x, y ∈ A : xRy ⇐⇒ ∃n ∈ Z ; x/y= 3^n


How can i show that is an equivalence relation (reflexivity symmetry,transitivity) on A ? And what is the equivalence classes of A?

I'm stuck on this 2 questions

Thanks!!

 

[pka]

Hello, Can someone help me with this problem?
I have a set :A={1,1/3,1/27,1/4,3,1/36,2,2/9,9/4,5}
The Relation is : ∀x, y ∈ A : xRy ⇐⇒ ∃n ∈ Z ; x/y= 3^n
How can i show that is an equivalence relation (reflexivity symmetry,transitivity) on A ? And what is the equivalence classes of A?

I will not do these for you. But here are strong hints.
Is it the case that \(x/x=3^0~?\) If so does that mean \(x\mathcal{R}x~?\)
Is it the case that if \(x/y=3^j\) then \(y/x=3^{-j}~?\) What does that mean for symmetry?
If \(x/y=3^k~\&~y/z=3^j\) what can you say about \(x/z~?\) So what?

 

[Dr.Peterson]

I have a set :
A={1,1/3,1/27,1/4,3,1/36,2,2/9,9/4,5}

The Relation is :

∀x, y ∈ A : xRy ⇐⇒ ∃n ∈ Z ; x/y= 3^n

How can i show that is an equivalence relation (reflexivity symmetry,transitivity) on A ? And what is the equivalence classes of A?

In addition to pka's very strong hints, since this relation is applied to a finite set, you will need to consider that specific set, especially when it comes to equivalence classes. I would probably list set A in order and make a table of which elements are related. This might also help you better understand the relation itself and see some of its properties.

 

[pka]

In addition to pka's very strong hints

I have taken note of your kind remark, thank you. But truth be told this is not the first time I have encounter this relation.
I tried to find my notes on this to no-avail. I do remember editing a similar question. I liked the relation.
However, it only works on a set of positive rational numbers.
Therefore, I see no reason why the set \(A\) is the domain here???