Equivalence Relation, prove dom(R) = range(R) = X

In summary: And since R is an equivalence relation, if <a,b> is in R, then <b,a> is also in R. So for every element 'a' in X, there exists an element 'b' in X such that <a,b> is in R, and vice versa. Therefore, the range of R is equal to X.
  • #1
Ceci020
11
0

Homework Statement


Given:
R is an equivalence relation over a nonempty set X

Prove:
dom(R) = X
and range(R) = X

Homework Equations

The Attempt at a Solution


I have the following thoughts:
About the given:
Since R is an equivalence relation over X by hypothesis, R satisfies:
Reflexivity: <x,x> belongs to R
Symmetry: <x,y> belongs to R, and <y,x> belongs to R
Transitivity: <x,y> belongs to R, <y,z> belongs to R, and <x,z> belongs to R

with x, y, z E X

About the conclusion:
Base on definition of domain and range of a relation R over a set X, I have:
dom(R) = {x E X : there exists y belongs to Y such that <x,y> E R}
range(R) = {y E Y : there exists x belongs to X such that <x,y> E R} What I'm confused is that I don't know how to connect my ideas together. The properties that R satisfies is with x, y, and z E X. And R is a subset of X x X. There is no Y whatsoever. So what should I do (or say) next to come to the conclusion?Thank you for your help.
 
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  • #2
Ceci020 said:

Homework Statement


Given:
R is an equivalence relation over a nonempty set X

Prove:
dom(R) = X
and range(R) = X


Homework Equations




The Attempt at a Solution


I have the following thoughts:
About the given:
Since R is an equivalence relation over X by hypothesis, R satisfies:
Reflexivity: <x,x> belongs to R
Symmetry: <x,y> belongs to R, and <y,x> belongs to R
Transitivity: <x,y> belongs to R, <y,z> belongs to R, and <x,z> belongs to R

with x, y, z E X

About the conclusion:
Base on definition of domain and range of a relation R over a set X, I have:
dom(R) = {x E X : there exists y belongs to Y such that <x,y> E R}
range(R) = {y E Y : there exists x belongs to X such that <x,y> E R}


What I'm confused is that I don't know how to connect my ideas together. The properties that R satisfies is with x, y, and z E X. And R is a subset of X x X. There is no Y whatsoever. So what should I do (or say) next to come to the conclusion?


Thank you for your help.

Reflexive is ALL you need. <x,x> is in R for ALL x in X, yes?
 
  • #3
I'm sorry but can you explain a little bit more?

I think that since for all x in X, <x,x> is always in R holds, then according to the conditions of the domain, there indeed exists x E X such that there also exists a Y (namely, y = x) that makes <x,y> E R. But how about the range?

thank you
 
  • #4
Ceci020 said:
I'm sorry but can you explain a little bit more?

I think that since for all x in X, <x,x> is always in R holds, then according to the conditions of the domain, there indeed exists x E X such that there also exists a Y (namely, y = x) that makes <x,y> E R. But how about the range?

thank you

If <a,b> is in R, then 'a' is in the domain and 'b' is in the range, right?
 

FAQ: Equivalence Relation, prove dom(R) = range(R) = X

1. What is an equivalence relation?

An equivalence relation is a relation on a set that is reflexive, symmetric, and transitive. This means that for any elements a, b, and c in the set, the relation satisfies the following properties: a is related to itself (reflexive), if a is related to b then b is related to a (symmetric), and if a is related to b and b is related to c, then a is related to c (transitive).

2. How do you prove that dom(R) = range(R) = X?

To prove that dom(R) = range(R) = X, you need to show that every element in X is both a domain and a range of R. This can be done by showing that for every x in X, there exists an element y in X such that (x,y) is in R and vice versa.

3. What is the significance of proving that dom(R) = range(R) = X?

This equivalence relation shows that every element in the set X is related to itself as well as to every other element in X. This can be useful in proving other mathematical properties and can also help to define partitions of the set X.

4. Can you give an example of an equivalence relation where dom(R) = range(R) = X?

One example of an equivalence relation where dom(R) = range(R) = X is the relation "has the same age as" on the set of all people. This relation is reflexive, symmetric, and transitive, and every person in the set is related to themselves and to every other person in the set.

5. Are there any other properties that an equivalence relation must satisfy?

Yes, there are two additional properties that an equivalence relation must satisfy: it must be well-defined and it must be connected. Well-defined means that the relation must be defined in a way that is consistent and unambiguous. Connected means that for any two elements in the set, there exists a sequence of elements that are related to each other.

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