It breaks the rule number of elements of a powerset of Set A = 2^n . Also contradicts null set is a subset of everyset. Also some places it says ∅ is a element of every set

2^4 = 16, but my results say 39

Example for set {1,2,3,4}

There can be 5 no. of sets n+1

single element = n

2 element = 10 - {1,2} {1,3} {1,4} {2,3} {2,4} {3,4} {1,∅} {2,∅} (3,∅) (4,∅)

3 elements = 10

4 elemets = 4 =>{1,2,3,4} {1,2,3,∅} {1,2,∅,4} {1,∅,3,4} {∅,2,3,4}

5 element = 5

additional the ∅

Edited Friday at 07:57 PM1 day by HbWhi5F

1 hour ago, HbWhi5F said:

some places it says ∅ is a element of every set

Which places?

3 hours ago, HbWhi5F said:

It breaks the rule number of elements of a powerset of Set A = 2^n . Also contradicts null set is a subset of everyset. Also some places it says ∅ is a element of every set

2^4 = 16, but my results say 39

Example for set {1,2,3,4}

There can be 5 no. of sets n+1

single element = n

2 element = 10 - {1,2} {1,3} {1,4} {2,3} {2,4} {3,4} {1,∅} {2,∅} (3,∅) (4,∅)

3 elements = 10

4 elemets = 4 =>{1,2,3,4} {1,2,3,∅} {1,2,∅,4} {1,∅,3,4} {∅,2,3,4}

5 element = 5

additional the ∅

Since you have set out your working I will answer this question for you, using you definition.

Your example set is defined by the list {1,2,3,4} that is 4 members so n = 4 and 2ⁿ = 16

Note 1,2,3,4 are numbers not sets or subsets, but these are the members of your set.

3 hours ago, HbWhi5F said:

Also some places it says ∅ is a element of every set

I very much doubt it. What it should say if written properly is that ∅ is a subset of every set - not a member.

It should also say that the Power set is the number of subsets that can be formed from the members, including the whole set itself.

So let us count up , remembering that ∅ is a subset, but not a member and that {3,4} and {4,3} refer to the same subset - order of members doesn't matter.

There is 1 subset with zero members - ∅

There is 1 subset with 4 members {1,2,3,4}

There are 4 subsets with 1 member

There are 4 subsets with 3 members

There are 6 subsets with 2 members.

Since this is posted in Homework help I will leave you to list the subsets I did not.

They are easy to check if you wish to post them.

33 minutes ago, studiot said:

members

You call them members; we call them elements.

9 hours ago, Genady said:

You call them members; we call them elements.

Who are we ?

Cantor's original word in German was indeed Elemente (plural).

German, like English received the word element via Latin from Greek, and the word has has many different meanings through the millenia.

Workers after Cantor discovered philosophical difficulties underlying basic set theory, which is possibly why Russell used the translation 'members' to distinguish aggregates (the English word for set at that time) which sets which contain other sets as members and sets which do not.

Whether you chose to use the word elements or members, the important point is to distinguish between subsets, which are neither elements nor members, and the elements/members themselves of the set.

Subsets are entites that are formed from the elements/members but are not generally members, which is why HbWhi5F has counted too many 'elements'.

That way the definitions are self consistent.

Edited 14 hours ago14 hr by studiot

3 minutes ago, studiot said:

Subsets are entites that are formed from the elements/members but are not generally members

Moreover, subsets of a set are never elements of that set.

5 minutes ago, studiot said:

which is why HbWhi5F has counted too many 'elements'

They are heavily confused.

1 minute ago, Genady said:

Moreover, subsets of a set are never elements of that set.

The set of everything that is not a square root ?

6 minutes ago, studiot said:

The set of everything that is not a square root ?

"Everything" is not a set.

32 minutes ago, studiot said:

Who are we ?

Just from curiosity, I've just checked the "classics", "A Book of Abstract Algebra" by Charles C. Pinter. The word member appears there 28 times. The word element, 500.

21 minutes ago, Genady said:

"Everything" is not a set.

Yes my wording was a bit slack.

But the set of every thing is most definitely an infinite set.

So if you restrict things to those without a square root, that set does not have a square root and must be a member of itself.

That is exactly why Russell introduced 'type' theory. also originally called category theory, but category theory now refers to a different subject.

1 minute ago, studiot said:

set of every thing

This is not a set either.

1 minute ago, studiot said:

Russell

Zermelo-Fraenkel

11 minutes ago, Genady said:

This is not a set either.

Why not ?

Thing is a perfectly acceptable general noun.

I also think that if you wish to continue this side discussion it would be better done in another thread, so as not to confuse a beginner in set theory.

In my view the standard introductory approach leaves much to be desired as it avoids introducing concepts that make set theory useful in so many other disciplines.

1 minute ago, studiot said:

Why not ?

You've just proved by contradiction that it is not a set: if it is a set then it is an element of itself, but it follows from Zermelo-Fraenkel axioms that no set can be element of itself.

See, e.g., Set is Not Element of Itself - ProofWiki

8 minutes ago, Genady said:

You've just proved by contradiction that it is not a set: if it is a set then it is an element of itself, but it follows from Zermelo-Fraenkel axioms that no set can be element of itself.

See, e.g., Set is Not Element of Itself - ProofWiki

You can't prove a definition.

And there are many alternative axiomatic systems of set theory, but none are complete because of the underlying tensions identified by Godel and others.