Understanding the Big Union Notation: Simple Examples and Explanation

[Pithikos]

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" .

I know the Summation formula and I could understand this Union formula if it didn't have that A_i that came from nowhere. Could someone please give a simple example on this one?

 

[discrete*]

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 $$A_{i}$$ indexed by $$i\in{I}$$." The union and intersections shown there are just the union -- or intersection -- of such $$A_{i}$$.

 

[Pithikos]

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 $$A_{i}$$ indexed by $$i\in{I}$$." The union and intersections shown there are just the union -- or intersection -- of such $$A_{i}$$.

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:

$$\bigcup^{}_{i \in I}$$A_i=A₁$$\cup$$A₂$$\cup$$A₃$$\cup$$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.

 

[discrete*]

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:

$$\bigcup^{}_{i \in I}$$A_i=A₁$$\cup$$A₂$$\cup$$A₃$$\cup$$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?

 

[disregardthat]

Yes, it is implicit that the set $$\{A_i\}_{i \in I}$$, which is the set of A_i for any i in I must be defined before you take the union of them $$\bigcup_{i \in I}A_i$$.

 

[Hurkyl]

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:

$$\bigcup^{}_{i \in I}$$A_i=A₁$$\cup$$A₂$$\cup$$A₃$$\cup$$A₄

This formula is the union of teh A_i's. If the A_i's aren't the sets you want to union, then this formula won't compute their union.

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 A_i=i. Or, you could forgo temporary variables entirely and write:

$$\bigcup_{x\in \{Q,W,E,R\}} x$$​

 

[Pithikos]

Ok thanks! I got it