Set Theory , building a set notation

[reenmachine]

Homework Statement

Build a set notation for $$\bigcup_{i \in N}R × [i , i + 1]$$

The Attempt at a Solution

##\{(x,y) \in R : x \in R \ \ \exists z \in N \ \ z ≤ y ≤ (z+1)\}##

Last time I tried one of these kind of sets I struggled quite a bit , so I'm interested in knowing how much I can handle those from now on.

thanks!

 

[reenmachine]

Take note that I read the notation as $$\bigcup_{i \in N}(R × [i , i +1])$$

 

[ArcanaNoir]

This isn't quite right. Can you visualize the elements in the union? Can you describe them with words? When you put [itex] (x,y) \in \mathbb{R}[/itex], you are describing intervals on the real line, is that what you're going for? It looks to me like you should be describing something in 2 dimensions, not 1 dimension.

 

[reenmachine]

I knew something looked fishy , seems I didn't learn from my mistakes.

Seems to me that the union of all elements of this set will give us the set R.

(Since [i , i+1] with ##i \in N## will always be in R.)

 

[ArcanaNoir]

I knew something looked fishy , seems I didn't learn from my mistakes.

Seems to me that the union of all elements of this set will give us the set R.

(Since [i , i+1] with ##i \in N## will always be in R.)

Good point, I hadn't thought of it as the union, I was looking at individual elements. But remember, RxR is not the same as R. Is the difference clear?

 

[reenmachine]

Good point, I hadn't thought of it as the union, I was looking at individual elements. But remember, RxR is not the same as R. Is the difference clear?

Not sure.

My understanding of it is that ##R## will have elements in the form of ##r## while ##R × R## will have elements in the form of ##(r,r)##.But my point was that if you unionize ##R## or ##R × R## , you will end up with the same elements if you ""deconstruct"" the ordered pairs that are made of ##r## in ##(r,r)##.

The confusion is probably why should we called them componants? Can a number be a componant as well as an element of a set (the ordered pair as a set)?

 

[reenmachine]

I will attempt to express what I mean in a clearer way.

Let's take the set ##R × [i , i+1]## with ##i \in N##.We accept that all numbers between ##i## and ##i+1## including both will be elements of ##R## as well.##R## in ##R × [i , i+1]## means that the first member of the ordered pairs will be any real number.So the ordered pair would look like ##(x,y)## with ##x## being any real number and ##y## being any positive real number except maybe all the numbers between ##0## and ##1## if you think ##i ≠ 0## because ##0 ∉ N##.Then if you unionize the set ##R × [i , i+1]## , you will something like ##\{...(x,y) \cup (x,y) \cup (x,y)...\}##.Since you unionize the ordered pairs , it will give you the set of all elements of those unionized sets(ordered pairs in that case) , therefore will give you all elements of ##R## from all possible ##x## in##(x,y)##.

 

[reenmachine]

Another attempt :

##\{\{x\},\{x,y\} |\ x,y \in R \ \ y ≥ 1 \}##

EDIT: I finally checked the book's solution , which is ##\{(x,y) : x,y \in R \ \ y ≥ 1\}##

I find it weird that the dummy variables would be put between "()" like it's an ordered pair to present the set such as "the set of all (x,y) such that...".Anybody can enlighten me on using my method versus the one in the book concerning the left side of the notation

 

[christoff]

They're essentially equivalent, at least insofar as Kuratowski definition of ordered pairs is concerned. In this definition, the cartesian product of two sets [itex]X[/itex] and [itex]Y[/itex], denoted by [itex]X\times Y[/itex] is defined by the set of all ordered pairs [itex](x,y)=\{\{x\},\{x,y\}\}[/itex], where the object on the left is just a convenient shorthand for the set on the right. See http://en.wikipedia.org/wiki/Cartesian_product for more information.