Prove Q is the equivalence relation on A

In summary: It means that the domain and range of f are the same as the domains and ranges of R.In summary, Q is an equivalence relation on A that is reflexive, symmetric and transitive.
  • #1
Hirman
5
0
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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  • #2
what is equivalence relation by definition?
 
  • #3
R is an equivalence relation on R iff R is a binary relation on A that is reflexive on A, symmetric and transitive
 
  • #4
Hirman said:
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?
 
  • #5
PeroK said:
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)
 
  • #6
Hirman said:
is a binary relation on A
what is arbitrary relation on A
 
  • #7
Hirman said:
reflexive on A, symmetric and transitive
write the formulas
 
  • #8
PeroK said:
What does mean?
I think that it is not a good idea to use notation for reals
 
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  • #9
Hirman said:
f(x)∈ dom(R), f(y)∈ ran(R) and f(x)Rf(y)
Ah! I missed that. So what's the difficulty here?
 
  • #10
How to prove Q is an equivalence relation on A?
 
  • #11
Hirman said:
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.
 
  • #12
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.
 
  • #13
Hirman said:
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:

Hirman said:
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.
 
  • #14
Hirman said:
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)##.
 

Related to Prove Q is the equivalence relation on A

1. What is an equivalence relation?

An equivalence relation is a mathematical concept that defines a relationship between elements in a set. It is a binary relation that is reflexive, symmetric, and transitive.

2. How do you prove that Q is an equivalence relation?

To prove that Q is an equivalence relation on a set A, you must show that it satisfies the three properties: reflexivity, symmetry, and transitivity. This can be done by showing that for any elements a, b, and c in set A, a is related to itself (reflexivity), if a is related to b, then b is related to a (symmetry), and if a is related to b and b is related to c, then a is related to c (transitivity).

3. What is the importance of equivalence relations in mathematics?

Equivalence relations are important in mathematics because they allow us to classify elements in a set into distinct groups based on their relationships. This can help us better understand and analyze mathematical structures and systems.

4. Can you provide an example of an equivalence relation?

One example of an equivalence relation is the relation "is congruent to" on the set of triangles. This relation satisfies the three properties and allows us to classify triangles into different types (equilateral, isosceles, scalene) based on their side lengths.

5. How are equivalence relations used in real-world applications?

Equivalence relations have various real-world applications, such as in computer science for data clustering and in social sciences for grouping individuals based on shared characteristics. They are also used in statistics for creating equivalence classes to analyze data and in engineering for creating equivalence classes of components for reliability analysis.

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