Unions and intersections of collections of sets

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In summary, the notation is confusing, but the definitions make sense. The collection is the set of all numbers in a set, and the unions and intersections are the sets of all numbers that are in either of two sets.
  • #1
1MileCrash
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My proof class just took a turn for the worst for me - I don't understand this.

First, the notation is extremely confusing to me, I need help to make sure I'm getting this.

If An is some set for some natural number n such as [-n, n].

Then (script A) the collection is the set of all An? Is that correct?

Just a basis needed..


Now, the definitions of unions and intersections got me super confused. But what I am getting out of it.. is that

U(script A)

Is the set of all x that are in any of the An in the collection, while

(intersection) An is the set of all x that are in every An in the collection?

So, in my example,

U(script A) is the set of all x that are in at least one of the An, which is all real numbers, because all real numbers will fall into one of those intervals.

while

(intersection)(script A) is the set of all x that are in all An, which is the empty set, because no real number will fall into every one of those intervals.



Ugh.. does anyone even know what I'm talking about? This is strange.
 
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  • #2
1MileCrash said:
My proof class just took a turn for the worst for me - I don't understand this.

First, the notation is extremely confusing to me, I need help to make sure I'm getting this.

If An is some set for some natural number n such as [-n, n].

Then (script A) the collection is the set of all An? Is that correct?

We don't know how you defined [itex]\mathcal{A}[/itex]. But that definition makes sense so I think you are correct.

Just a basis needed..


Now, the definitions of unions and intersections got me super confused. But what I am getting out of it.. is that

U(script A)

Is the set of all x that are in any of the An in the collection, while

(intersection) An is the set of all x that are in every An in the collection?

Correct.

So, in my example,

U(script A) is the set of all x that are in at least one of the An, which is all real numbers, because all real numbers will fall into one of those intervals.

Correct.

while

(intersection)(script A) is the set of all x that are in all An, which is the empty set, because no real number will fall into every one of those intervals.

Not correct. The number 0 will be in every one of those intervals. So [itex]\bigcap \mathcal{A}=\{0\}[/itex]

Ugh.. does anyone even know what I'm talking about? This is strange.

It's weird notation, I know. But you will eventually get used to it. You seem to grasp it alright.
 
  • #3
Thank you.. I see my error. I think I confused it with this one here:

Consider An = { k >= n }

Where k is a natural number.

Would it be correct to say that the union of the collection is all real numbers, since all k can fall into (at least one) An set?

And that the intersection is the empty set, since no natural number k is greater than or equal to ALL natural numbers n?
 
  • #4
1MileCrash said:
Thank you.. I see my error. I think I confused it with this one here:

Consider An = { k >= n }

Where k is a natural number.

I don't like that notation. You should write it better. For example

[tex]A_n=\{k\in \mathbb{N}~\vert~k\geq n\}[/tex]

Would it be correct to say that the union of the collection is all real numbers, since all k can fall into (at least one) An set?

But k are natural numbers. So you can't get all the real numbers. The union would be the set of all the natural numbers.

And that the intersection is the empty set, since no natural number k is greater than or equal to ALL natural numbers n?

Correct.
 
  • #5
Right right... just need to shut up and practice for now!

Thanks a bunch for your help again!
 

1. What is the union of two sets?

The union of two sets is a new set that contains all the elements from both sets. In other words, it combines the elements from both sets without any duplicates. It is denoted by the symbol "∪".

2. What is the intersection of two sets?

The intersection of two sets is a new set that contains only the elements that are common to both sets. In other words, it is the set of elements that are present in both sets. It is denoted by the symbol "∩".

3. How is the union of more than two sets calculated?

The union of more than two sets is calculated by combining all the elements from each set and removing any duplicates. This can be represented using the union symbol "∪" and the sets can be listed inside parentheses, for example: A ∪ (B ∪ C).

4. How is the intersection of more than two sets calculated?

The intersection of more than two sets is calculated by finding the common elements between all the sets. This can be represented using the intersection symbol "∩" and the sets can be listed inside parentheses, for example: A ∩ (B ∩ C).

5. What is the difference between union and intersection?

The union and intersection of two sets are opposite operations. The union combines elements from both sets, while the intersection only includes elements that are common to both sets. In other words, the union expands the set, while the intersection narrows it down.

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