[QUOTE=science_man_88;249944][TEX]\empty[/TEX][/QUOTE]

That's a subset that every set has, It's not a set with an element which is also a subset. It's a set with no elements at all.

[QUOTE=Mr. P-1;249947]That's a subset that every set has, It's not a set with an element which is also a subset. It's a set with no elements at all.[/QUOTE]

then your hint lead me to nothing because it fits you hint of a subset that every set has. the only way i see your hint working is if you see the empty set as a collection of empty sets which means the empty set is a element of the collection we call the empty set.

The question is to find a set S with
1. [TEX]s\in S[/TEX], and
2. [TEX]s\subseteq S[/TEX]
for some s.

For example, {2, 3, 5, 7} is a subset (with four elements) of {0, 1, 2, 3, 4, 5, 6, 7, 8, 9} and 6 is an element of that set. {1, 2, {1, 2, 3, 4}} is a subset (with three elements) of {1, 2, 3, {1, 2, 3, 4}}, while {1, 2, 3, 4} is an element. But none of the subsets in my examples are elements (members), and none of the elements are subsets.

[QUOTE=Mr. P-1;249942]
Please give an example of a set with an element which is also a subset. Hint: What subset does every set have?[/QUOTE]

the only subset every set has has been said to be [TEX]\empty[/TEX] so he's hinting that [TEX]\empty[/TEX] is the answer but then claiming it's not.

[QUOTE=science_man_88;249962]the only subset every set has has been said to be [TEX]\empty[/TEX] so he's hinting that [TEX]\empty[/TEX] is the answer but then claiming it's not.[/QUOTE]

You're answering a question he isn't asking.

You can freely ignore the hint, if you like. It gives one right answer (which you haven't found yet) but there are others.

[QUOTE=CRGreathouse;249968]You're answering a question he isn't asking.

You can freely ignore the hint, if you like. It gives one right answer (which you haven't found yet) but there are others.[/QUOTE]

I'm obviously not getting it at all from the question all i can think of is every 1 element set and from his hint the only possibilities I've seen are [TEX]\empty[/TEX] and the set itself.

[QUOTE=science_man_88;249973]I'm obviously not getting it at all from the question all i can think of is every 1 element set and from his hint the only possibilities I've seen are [TEX]\empty[/TEX] and the set itself.[/QUOTE]

The set S itself isn't a member of S -- ZF doesn't allow that, so that won't work. The empty set is a subset of S since it's a subset of everything, but is it an element of S? (Can you choose an S that makes that true?)

[QUOTE=CRGreathouse;249982]The set S itself isn't a member of S -- ZF doesn't allow that, so that won't work. The empty set is a subset of S since it's a subset of everything, but is it an element of S? (Can you choose an S that makes that true?)[/QUOTE]

obviously not as you wouldn't have to ask then.

[QUOTE=science_man_88;249994]obviously not as you wouldn't have to ask then.[/QUOTE]

Sure I would have asked. The possibilities are you can (in which case give an example) or you can't (in which case give a reason).

[QUOTE=CRGreathouse;249998]Sure I would have asked. The possibilities are you can (in which case give an example) or you can't (in which case give a reason).[/QUOTE]

well if a set can have one element an element is a subset of a set one element in length for all sets that I can think of ( maybe one exception). a set is a collection of individual elements, a subset is a set that has at least one of those elements in it, by that logic 1 element from any set can be a subset.

[QUOTE=science_man_88;249973]I'm obviously not getting it at all from the question all i can think of is every 1 element set and from his hint the only possibilities I've seen are [TEX]\empty[/TEX] and the set itself.[/QUOTE]

You're almost there.

If A is a 1-element set, then the only subsets of A are [TEX]\empty[/TEX] and A itself.

A cannot be an element of A. What could be an element of A?