# 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.

## Answers and Replies

Science Advisor
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

Science Advisor
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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)

Science Advisor
is a binary relation on A
what is arbitrary relation on A

Science Advisor
reflexive on A, symmetric and transitive
write the formulas

Science Advisor
What does mean?
I think that it is not a good idea to use notation for reals

Delta2
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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?

Science Advisor
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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.

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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.

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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)##.