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.
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.
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.
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.]
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.]
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.