Confusion about empty set and singleton sets as elements

[gflores]

Need some help on some of these problems. Thanks in advanced.

Ø
E = is an element of...

1. Ø E {{Ø}1,2}
False? I'm a little confused. Is Ø on the left side a set? So, can it be equivalently written as {Ø}? If so, then it can't be an element of something, since it's actually a set, or in this case a subset.
2. {2} E {1,2,3}
Confused again here with the 2 being in braces, does it mean it's a set and therefore can't be an element of something? False?
3. Ø = {Ø}
No idea here. I'm guessing true.

Any help is appreciated. Thanks again.

 

[Muzza]

Why can't a set be an element of another set?

2) is indeed false, but not for the reason you gave. {2} is not an element of {1,2,3} (if it were, {2} would be equal to 1, 2, or 3. A set equal to a natural number? Surely a contradiction). But {2} IS an element of {1,{2},3}.

{Ø} is a set of sets, while Ø itself is "just" a set. {Ø} has one member, but Ø has zero members.

An analogy: imagine that sets are "baskets". Imagine a really large basket, in which you could put other baskets (these smaller baskets may or may not contain other things). Then you have a "set" which holds other "sets".