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

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In summary: The conversation discusses the concept of sets and how they can be elements of other sets. It is explained that while a set may contain individual elements, it can also contain other sets as elements. This is compared to a shopping bag containing an apple and a package of sausages, where the sausages can only be accessed through the package. The concept of the null set is also discussed, with the example of a shopping bag containing only a package of sausages and no other items. The explanation mentions the difference between "naive set theory" and "class theory", where sets can be contained in other sets in the former but not the latter. In summary, the conversation discusses the possibility of a set being an element of another set and
  • #1
Venomily
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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
 

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  • #2
im quite desperate.
 
  • #3
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!
 
  • #4
2(d) is the null set. Try it with the package concept!
 
  • #5
UltrafastPED said:
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.
 
  • #6
{} = 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.
 
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  • #7
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.
 

1. How can an element be a part of a set?

In chemistry, an element is defined as a substance that cannot be broken down into simpler substances by chemical means. A set is a collection of distinct objects. Therefore, an element can be a part of a set if it meets the criteria of being a distinct object in that set.

2. What is the difference between an element and a set in science?

An element is a substance that is made up of only one type of atom, while a set is a collection of distinct objects. In other words, an element is a single entity, while a set is a group of entities.

3. Can an element be a part of multiple sets?

Yes, an element can be a part of multiple sets. In chemistry, an element can have different properties and characteristics that may make it a part of different sets. For example, oxygen can be a part of the set of non-metals as well as the set of gases.

4. How are elements and sets related in the periodic table?

The periodic table is a way of organizing elements based on their atomic structure and properties. Elements are placed in specific groups and periods in the periodic table based on their similarities and differences. Therefore, elements and sets are related in the periodic table as elements can be grouped into sets based on their properties.

5. Can an element be removed from a set?

Yes, an element can be removed from a set. In science, a set is a collection of distinct objects, and if an object no longer meets the criteria of being a part of that set, it can be removed. For example, if an element undergoes a chemical reaction and is transformed into a different substance, it may no longer be considered a part of its original set.

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