# Prove Q is the equivalence relation on A

## 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.

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wrobel
what is equivalence relation by definition?

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 Helper
Gold Member
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?

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

Delta2
PeroK
Homework Helper
Gold Member
f(x)∈ dom(R), f(y)∈ ran(R) and f(x)Rf(y)
Ah! I missed that. So what's the difficulty here?

How to prove Q is an equivalence relation on A?

PeroK
Homework Helper
Gold Member
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.

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
Homework Helper
Gold Member
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
Staff Emeritus