# Function assigning each subset of R sum of its elements

• Jan 11th 2013, 01:58 AM
wilhelm
Function assigning each subset of R sum of its elements
Hi.

Could you give me a hint how to solve this problem?

Let $\displaystyle D:= \left\{E \subset \mathbb{R} | 0< card(E)< + \infty \right\}$.

$\displaystyle \phi : D \ni E \rightarrow \sum_{x \in E} \ x \in \mathbb{R}$

Check if $\displaystyle \phi$ is injective or surjective.
• Jan 11th 2013, 02:17 AM
emakarov
Re: Function assigning each subset of R sum of its elements
Quote:

Originally Posted by wilhelm
Well, I think that it can't be injective because $\displaystyle card (\mathcal{P}(\mathbb{R})) > card(\mathbb{R})$, so by the pigeonhole principle, there must to at least two subsets with the same sum of elements.

There are two mistakes here. First, D is the collection of finite nonempty subsets of $\displaystyle \mathbb{R}$, so $\displaystyle \mathrm{card}(D)=\mathrm{card}(\mathbb{R})$. Second, pigeonhole principle is not valid for infinite sets. You may have an injection from a infinite set into its proper subset; in fact, this is one of the definitions of an infinite set. The proof of the pigeonhole principle proceeds by induction on the cardinality of the domain, so it only applies when this cardinality is a natural number. You need to use transfinite induction to prove properties of infinite numbers (ordinals). It is probably instructive to see where the proof of the principle breaks down when one tries to use transfinite induction instead of regular one.

Speaking about the problem, you need to have an intuition about D. Can you give examples of sets in D? Both questions are trivial once you understand what D is.
• Jan 11th 2013, 03:05 AM
wilhelm
Re: Function assigning each subset of R sum of its elements
Thank you, I've already deleted that comment. Could we just say that for example $\displaystyle \phi (\left\{ x \right\})= \phi (\left\{ x, 0 \right\})= x$ and $\displaystyle \forall x \in \mathbb{R} \ \ \exists E \subset \mathbb{R} : \phi(E)=x$ for example $\displaystyle E = \left\{ x \right\}$?
Or is it oversimplified?
• Jan 11th 2013, 03:15 AM
emakarov
Re: Function assigning each subset of R sum of its elements
Quote:

Originally Posted by wilhelm
Thank you, I've already deleted that comment. Could we just say that for example $\displaystyle \phi (\left\{ x \right\})= \phi (\left\{ x, 0 \right\})= x$ and $\displaystyle \forall x \in \mathbb{R} \ \ \exists E \subset \mathbb{R} : \phi(E)=x$ for example $\displaystyle E = \left\{ x \right\}$?

Yes. Concerning injection, you need $\displaystyle x\ne0$, but since to disprove injection you need to find just one pair of arguments mapped to the same image, such x obviously exists.
• Jan 11th 2013, 03:35 AM
wilhelm
Re: Function assigning each subset of R sum of its elements
Thank you. Could you maybe help me with the linear algebra problem about detA=1995, too?