Thread: Set theory question

Could anyone help me with the following question:

Let R be a relation. Show that domR={x : ∃y ([x, y] ∈ R)}, is a set (where [x,y] is and ordered pair.
Also, let a and b be sets, Prove that there exists a set whose members are exactly the functions with domain a and codomain b

Many thanks.

Originally Posted by KSM08

Could anyone help me with the following question:

Let R be a relation. Show that domR={x : ∃y ([x, y] ∈ R)}, is a set (where [x,y] is and ordered pair.
Also, let a and b be sets, Prove that there exists a set whose members are exactly the functions with domain a and codomain b

Many thanks.

Are you sure you've stated the question precisely? A relation is defined on a set. Domain and range are normally associated with a function.

Could you possibly scan the exact problem in?

Originally Posted by oldguynewstudent

Are you sure you've stated the question precisely? A relation is defined on a set. Domain and range are normally associated with a function. Could you possibly scan the exact problem in?

Actually, we say that a function is a relation with more restrictions.
Therefore, we do define domains and codomains(images) for relations.
That said I do have concerns with this question.

Originally Posted by KSM08

Prove that there exists a set whose members are exactly the functions with domain a and codomain b

In your text material what is expected when asked to prove existence of a set?
Are you given a set of set axioms dealing with this?
If so, we don’t know what axioms you have to use.

Originally Posted by KSM08

Show that domR={x : ∃y ([x, y] ∈ R)}, is a set

Every x in dom(R) is a member of UUR. So the desired set is a subset of UUR, so we obtain the desired set by the axiom schema of separation. ['U' stands for the unary union operation.]

Originally Posted by KSM08

Prove that there exists a set whose members are exactly the functions with domain a and codomain b

I suppose that you've already proven the existence of Cartesian products. Then we observe that the desired set is a subset of P(aXb). So we obtain the desired set from the axiom schema of separation. ['P' stands for the power set operation and 'X' stands for the Cartesian product operation.]

Originally Posted by oldguynewstudent

Domain and range are normally associated with a function.

Yes, but also some authors define 'dom' and 'range' on sets in general.

dom(S) = {x | Ey <x y> e S}
range(S) = {y Ex <x y> e S}

whether S is a function, relation, or set of any kind whatsoever.

Why be so stubborn, why not learn to post in standard symbols? You can use LaTeX tags.
[tex] \text{dom}(S) = \left\{ {x|\left( {\exists y} \right)\left[ {\left( {x,y} \right) \in S} \right]} \right\}[/tex] gives .
It is just so much easier to read standard notation.
I can tell you,as an editor and a reviewer, that makes a difference in what gets published.