questions about elements in sets ...

I am studying what sets are ...
And what I have learned is that we have "sets" and in those sets there are "elements"

So a set can be defined with {}
And if a set has no elements this is an emty set = ∅

So a set can be defined with elements 1, 3, 6 = {1, 3, 6}

But if we have the following set:
{{1,2,{3},{4,5},6,{{7},8}},9}

How do we determine how many elements there are in this set?

I would say 9 ... (1, 2, 3, 4, 5, 6, 7, 8, 9)
Or is it 2? (6, 9)
Or is this 7? (5 sets that can be seen as elements and 2 numbers)

Thanks in advance ...

Quote:

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Originally Posted by iwan1981

But if we have the following set:
{{1,2,{3},{4,5},6,{{7},8}},9}
How do we determine how many elements there are in this set?

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There are two elements in that set.
A number, 9 and a set {1,2,{3},{4,5},6,{{7},8}}.

But in this set .... {1,2,{3},{4,5},6,{{7},8}}
You can take more elements right?

Like this:

SET = {{1,2,{3},{4,5},6,{{7},8}},9}
- Element 1 = set {1,2,{3},{4,5},6,{{7},8}}
- Element 2 = number 9

The set that is actually an element = {1,2,{3},{4,5},6,{{7},8}}
- Element 1 = set {3}
- Element 2 = set {4, 5}
- Element 3 = set {{7}, 8}
- Element 4 = number 1
- Element 5 = number 2
- Element 6 = number 6

This set also had an element that has multiple sets = {{7}, 8}
- Element 1 = {7}
- Element 2 = number 8

So if we count all the elements the answer is 10 elements in total or am I wrong here ...

I guess that this has to do with how the question is asked ...
- What are is the total of elements that you can get from the following set?
- How many elements can you get from this set?

Right?

Quote:

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Originally Posted by iwan1981

But in this set .... {1,2,{3},{4,5},6,{{7},8}}

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In the set
There are six elements in the set :
1, 2 , {3}, {4,5}, 6 , {{7},8}.
That is three numbers and three sets.

Note that but .

just as I written here:

The set that is actually an element = {1,2,{3},{4,5},6,{{7},8}}
- Element 1 = set {3}
- Element 2 = set {4, 5}
- Element 3 = set {{7}, 8}
- Element 4 = number 1
- Element 5 = number 2
- Element 6 = number 6

SO 6 is an element of A but 3 is not because this is a set on it's own.

Quote:

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SET = {{1,2,{3},{4,5},6,{{7},8}},9}...
So if we count all the elements the answer is 10 elements in total or am I wrong here ...

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When we are talking about elements of a set, we are only talking about immediate elements, not nested elements. Therefore, the original set {{1,2,{3},{4,5},6,{{7},8}},9} has only two elements.

This is like having physical folders that can contain other folders as well as individual sheets of paper. Suppose folders are opaque. Then the number of elements of a folder F is how many physical objects (other folders and sheets) you see when you open F, without opening any other folders.

Quote:

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I guess that this has to do with how the question is asked ...
- What are is the total of elements that you can get from the following set?
- How many elements can you get from this set?

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People usually don't talk about elements that "one gets" from a set. The standard concept is an element of a set. One first has to define precisely what are elements "one gets" from a set: they are probably members of the transitive closure of the set.