Trouble Understanding Indexed Union of Lines in the Plane

In summary, the conversation is about the indexed union of In, which is a subset of R2 and is defined as the line of equation y=n. There is confusion about how the indexed union relates to a subset of the plane, and it is clarified that the individual sets of the indexed union each contain a line and all the points that make up the line. The conversation ends with the understanding that the indexed union is not an intersection and the individual sets contain different lines.
  • #1
MotoPayton
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I am having trouble understanding how the indexed union of ln in the first picture is equal to a subset of the plane; an element of it is a point on one of the lines. If I were to choose say 0 1 2 then the indexed union should be y=0 union y=1 union y=2. These lines would have no points in common so the total indexed union should be the empty set. I understand the second second set, it is the indexed union giving me trouble. Sorry if the pictures are blurry my phone isn't the greatest.

0306132054a.jpg


http://i603.photobucket.com/albums/tt113/KtmPayton/0306132054.jpg
 
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  • #2
MotoPayton said:
These lines would have no points in common so the total indexed union should be the empty set.


You didn't include the part of the text where [itex] I_n [/itex] is defined.

Why would having points in common be relevant to a union of sets? Are you thinking of an intersection of sets instead?
 
  • #3
The definition of In is on the next link. In is a subset of R2 and is the line of equation y=n.
I have to admit for some reason I was stuck on thinking this was an intersection. Thinking about it some more I have realized that the individual sets of the indexed union each contain a line. Those particular sets contain all the points that make up the line.
That is how it differs from the other one where the individual set contains all the different lines. Pretty sure I have it now. Thanks for helping.
 

1. What is the indexed union of lines in the plane?

The indexed union of lines in the plane is a mathematical concept that represents the collection of all lines in the plane, each with a unique index or label. It is denoted by the symbol ⋃ᵢ Lᵢ, where i represents the index of the line.

2. How is the indexed union of lines in the plane different from a regular union of lines?

The indexed union of lines in the plane differs from a regular union of lines in that it includes all lines in the plane, each labeled with a unique index. This means that even if two lines have the same equation or properties, they will still be considered separate lines in the indexed union.

3. What is the purpose of using an indexed union of lines in the plane?

The purpose of using an indexed union of lines in the plane is to have a systematic way of representing all possible lines in the plane. By assigning each line a unique index, it becomes easier to identify and work with specific lines, especially when dealing with a large number of lines.

4. How can I visualize the indexed union of lines in the plane?

One way to visualize the indexed union of lines in the plane is to imagine a grid with an infinite number of lines passing through it. Each line is labeled with a unique index, and together they make up the indexed union of lines in the plane.

5. Can the indexed union of lines in the plane be extended to higher dimensions?

Yes, the concept of indexed union of lines can be extended to higher dimensions, such as the indexed union of planes in 3-dimensional space. The same principle applies, where each plane is assigned a unique index and together they make up the indexed union.

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