# Question about element of and subset symbols

1. Oct 15, 2011

### fishingspree2

Question about "element of" and "subset" symbols

I've always thought that ∈ is defined when talking about elements in a set. For example, if A is a set and x is an element, then x ∈ A is defined. It wouldn't make sense to say x $\subseteq$ A

In the same way, if A and B are sets and A is contained in B, then it is incorrect to say A ∈ B. We should use A$\subseteq$ B.

Question is: say a set which contains the empty set: {Ø}
I would think we should write Ø $\subseteq$ {Ø} because Ø is a set itself.
But at the same time Ø ∈ {Ø} looks like it could also make sense... I am slightly confused.

Can anyone shed some light on the matter? Thank you very much.

2. Oct 16, 2011

### Stephen Tashi

Re: Question about "element of" and "subset" symbols

You go to the trouble of distinguishing between "element of" and "subset of" and then you use the ambiguous term "contains". Tisk, tisk.

3. Oct 16, 2011

### HallsofIvy

Staff Emeritus
Re: Question about "element of" and "subset" symbols

Specifically, fishingspree2, you are being ambiguous when you say "if A and B are sets and A is contained in B" without distinguishing the two meanings of "contained in". In "naive set theory" it is perfectly possible to have a set whose members are sets. Given that $\{\Phi\}$ is the set whose only member is the empty set, it is correct to say that the empty set is a member of that set. And, since the empty set is a subset of any set, both $\Phi\subset \{\Phi\}$ and $\Phi\in\{\Phi\}$ are both valid.

4. Oct 16, 2011

### disregardthat

Re: Question about "element of" and "subset" symbols

In set theory every element of a set is a set. There are no mathematical objects but sets in set theory.

5. Oct 26, 2011

### xxxx0xxxx

Re: Question about "element of" and "subset" symbols

Both are true, in the one case,
$$x \mbox{ is a set} \Rightarrow ( \emptyset \subseteq x)$$
and the other,
$$x \in \{ \emptyset \} \Leftrightarrow x = \emptyset$$.