What is the difference between sets and classes in set theory?

[Venomily]

Question 2 (a)

how is it possible? B is a set (since A is a set), how can a set be an element of another set?

Rather than saying: B is an element of C

I thought it would be better to say: B is a subset of C.



Also, can someone explain question 2 (d) to me? thanks

 

[Venomily]

im quite desperate.

 

[UltrafastPED]

Yes, a set can be an element of another set. There may be a set X={a,b,c}, where X is an element of Y.
But neither a, nor b, nor c becomes a member of Y; only the sets are members.

But you can have a set Z = {a, X} ... so X contains a,b,c, but neither b nor c is a member of Z.

Sets are "packages", and the set Z has the package X as an element - like having an apple and a package of sausages in your shopping bag. You can reach in and pull out the apple, but you cannot pull out a sausage - only a package of sausages.

But you could reach into that package of sausages and pull out a sausage!

 

[UltrafastPED]

2(d) is the null set. Try it with the package concept!

 

[Venomily]

Yes, a set can be an element of another set. There may be a set X={a,b,c}, where X is an element of Y.
But neither a, nor b, nor c becomes a member of Y; only the sets are members.

But you can have a set Z = {a, X} ... so X contains a,b,c, but neither b nor c is a member of Z.

Sets are "packages", and the set Z has the package X as an element - like having an apple and a package of sausages in your shopping bag. You can reach in and pull out the apple, but you cannot pull out a sausage - only a package of sausages.

But you could reach into that package of sausages and pull out a sausage!

Great explanation, I understand now, thanks.

But how would I go about 2 (d)? we have the empty set, {}:

{A} = { { 0, {}, {{}} } }

Judging by what you said, I don't see how {} can be related to the above set? it is neither an element nor a set.

 

[Venomily]

{} = sausages
{{}} = packet of sausages
0 = Orange

{ 0, {}, {{}} } = Shopping bag of (Orange + sausages + Packet of sausages).

{ { 0, {}, {{}} } } = car boot of Shopping bag.

We want {{}} i.e. the packet of sausages. It is not related to the Car boot because it is neither a SET nor a MEMBER of the Car boot, it is too deep inside.

 

[HallsofIvy]

By the way, this is true in "Naive set theory" which suffers from "Russel's Paradox". In "class theory" we do NOT allow "sets" to be contained in sets, but have a hierarchy of "classes" in which classes in one tier can be contained in classes of a higher tier. "Sets" are the lowest tier of classes.