Question about element of and subset symbols

[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.

 

[Stephen Tashi]

Question is: say a set which contains the empty set: {Ø}

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

 

[HallsofIvy]

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.

 

[disregardthat]

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

 

[xxxx0xxxx]

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.

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