Prove Q is the equivalence relation on A

[Hirman]

Homework Statement

Assume that f:A—>B and that R is an equivalence relation on B
Define Q to be the set {<x,y> ∈ A X A |<f(x),f(y)> ∈ R}

Relevant Equations

Define Q to be the set {<x,y> ∈ A X A |<f(x),f(y)> ∈ R}

I can’t understand it.

 

[wrobel]

what is equivalence relation by definition?

 

[Hirman]

R is an equivalence relation on R iff R is a binary relation on A that is reflexive on A, symmetric and transitive

 

[PeroK]

Homework Statement:: Assume that f:A—>B and that R is an equivalence relation on B
Define Q to be the set {<x,y> ∈ A X A |<f(x),f(y)> ∈ R}
Relevant Equations:: Define Q to be the set {<x,y> ∈ A X A |<f(x),f(y)> ∈ R}

I can’t understand it.

What does $\langle f(x), f(y) \rangle \in \mathbb R$ mean?

 

[Hirman]

What does $\langle f(x), f(y) \rangle \in \mathbb R$ mean?

f(x)∈ dom(R), f(y)∈ ran(R) and f(x)Rf(y)

 

[wrobel]

is a binary relation on A

what is arbitrary relation on A

 

[wrobel]

reflexive on A, symmetric and transitive

write the formulas

 

[wrobel]

What does mean?

I think that it is not a good idea to use notation for reals

 

[PeroK]

f(x)∈ dom(R), f(y)∈ ran(R) and f(x)Rf(y)

Ah! I missed that. So what's the difficulty here?

 

[Hirman]

How to prove Q is an equivalence relation on A?

 

[PeroK]

How to prove Q is an equivalence relation on A?

Okay, what exactly is stopping you getting started? The rules are you have to show an attempt at a solution.

 

[Hirman]

I’m sorry.But I have no ideal how to get start.I can't see the connection between x and y in A, the only connection is in B.But B is another set.

 

[PeroK]

I’m sorry.But I have no ideal how to get start.I can't see the connection between x and y in A, the only connection is in B.But B is another set.

You know this:

R is an equivalence relation on R iff R is a binary relation on A that is reflexive on A, symmetric and transitive

That's true for $Q$ as well, right? In fact, I would have written:

An equivalence relation (on a set) is a binary relation (on that set) that is reflexive, symmetric and transitive.

You need to prove that $Q$ is reflexive, symmetric and transitive. Okay? That's you started.

 

[vela]

f(x)∈ dom(R), f(y)∈ ran(R) and f(x)Rf(y)

That's not right. You're told f is a function from A to B, so x and y are in A = dom(f) while f(x) and f(y) are in B = ran(f).

$\langle f(x), f(y) \rangle \in R$ is just another way of writing $f(x) R f(y)$.