What is the big Union?

  • Thread starter Pithikos
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  • #1
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Main Question or Discussion Point

In my discrete math book there is half a page with very formal explanation of the big Union notation and two very short examples without guidance so I have a hard time understanding what goes on. Here's a http://img525.imageshack.us/img525/8507/unionl.jpg" [Broken].

I know the Summation formula and I could understand this Union formula if it didn't have that Ai that came from nowhere. Could someone please give a simple example on this one?
 
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  • #2
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This is a general definition of what we refer to as index sets. Indexes appear in all branches of mathematics, and you've undoubtedly seen them before. Many times where you see a "subscript", that's usually an indication that something is being indexed by the subscript.

The case here is not much different. The first definition (3.10) is read as "...the set [tex]A_{i}[/tex] indexed by [tex]i\in{I}[/tex]." The union and intersections shown there are just the union -- or intersection -- of such [tex]A_{i}[/tex].
 
  • #3
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This is a general definition of what we refer to as index sets. Indexes appear in all branches of mathematics, and you've undoubtedly seen them before. Many times where you see a "subscript", that's usually an indication that something is being indexed by the subscript.

The case here is not much different. The first definition (3.10) is read as "...the set [tex]A_{i}[/tex] indexed by [tex]i\in{I}[/tex]." The union and intersections shown there are just the union -- or intersection -- of such [tex]A_{i}[/tex].
I understand the index part. I just can't understand what that has to do with the union of some elements.

For example say that set I is {1,2,3,4}. Then I have Q={1, 3}, W={5, 7}, E={100,101}, R={5, 10} and I want to unite those together. If I apply the formula I would get:

[tex]\bigcup^{}_{i \in I}[/tex]Ai=A1[tex]\cup[/tex]A2[tex]\cup[/tex]A3[tex]\cup[/tex]A4

Which doesn't make sense to me as A1, A2.. are not defined anywhere. With my thinking this would work only if instead of Q, W, E and R, I used A1, A2, A3 and A4 when naming my sets.
 
  • #4
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I understand the index part. I just can't understand what that has to do with the union of some elements.

For example say that set I is {1,2,3,4}. Then I have Q={1, 3}, W={5, 7}, E={100,101}, R={5, 10} and I want to unite those together. If I apply the formula I would get:

[tex]\bigcup^{}_{i \in I}[/tex]Ai=A1[tex]\cup[/tex]A2[tex]\cup[/tex]A3[tex]\cup[/tex]A4

Which doesn't make sense to me as A1, A2.. are not defined anywhere. With my thinking this would work only if instead of Q, W, E and R, I used A1, A2, A3 and A4 when naming my sets.
You lost me. Why is A1, A2 not defined? And where/why are the other sets coming into play?
 
  • #5
disregardthat
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Yes, it is implicit that the set [tex]\{A_i\}_{i \in I}[/tex], which is the set of A_i for any i in I must be defined before you take the union of them [tex]\bigcup_{i \in I}A_i[/tex].
 
  • #6
Hurkyl
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I understand the index part. I just can't understand what that has to do with the union of some elements.

For example say that set I is {1,2,3,4}. Then I have Q={1, 3}, W={5, 7}, E={100,101}, R={5, 10} and I want to unite those together. If I apply the formula I would get:

[tex]\bigcup^{}_{i \in I}[/tex]Ai=A1[tex]\cup[/tex]A2[tex]\cup[/tex]A3[tex]\cup[/tex]A4
This formula is the union of teh Ai's. If the Ai's aren't the sets you want to union, then this formula won't compute their union. :tongue:

With my thinking this would work only if instead of Q, W, E and R, I used A1, A2, A3 and A4 when naming my sets.
Why "instead of"? You get to choose what I and what the Ai's are.

Incidentally, you could have instead used I = {Q,W,E,R} and set Ai=i. Or, you could forgo temporary variables entirely and write:
[tex]\bigcup_{x\in \{Q,W,E,R\}} x[/tex]​
 
  • #7
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Ok thanks! I got it :biggrin:
 

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