Union and Intersection of sets

[Goldenwind]

[SOLVED] Union and Intersection of sets

Homework Statement

a) Find: \bigcup_{i=1}^{\infty}} A_i


b) Find: \bigcap_{i=1}^{\infty}} A_i

Where A_i = (0,i), that is, the set of real numbers x with 0 < x < i

I was doing okay when they gave me A_i = {i, i+1, i+2, ...}, but now that they're giving me (0,i), and introducing x, I'm getting confused.

 

[Gib Z]

The new extension isn't that much harder =]

That just means A_i is the set of all the real numbers between 0 and i.

To do this problem, you may find it helpful to draw a real number line in the positive direction starting from 0, and draw a line above the first set, i=1, which would be a line from 0 to 1, non inclusive, then i=2..so on so on. Note that integers are never part of the sets.

 

[Goldenwind]

So the union would be... N?
And the intersection would be... the range (0,1)? (Since i starts at 1, all instances of this would include (0,1), right?)

^^;

 

[Gib Z]

Noooo! I'm so stupid! Sorry I made a silly mistake.

Ok the Union of the sets is the Set which has every element in the original sets.

The Intersection is the set which has only the elements possessed commonly by every set.

If we had followed by draw a number line plan, I would have stopped my error faster!
Sure A_1 does not include 1, and A_2 does not include 2, and A_3 does not include 3, but A_3 included 2 and 1, and A_22 won't include 9, but every positive integer less than 9.

The Union is Not N :( But The intersection is indeed every real number on the open interval (0,1).

 

[Goldenwind]

Noooo! I'm so stupid! Sorry I made a silly mistake.

Ok the Union of the sets is the Set which has every element in the original sets.

The Intersection is the set which has only the elements possessed commonly by every set.

If we had followed by draw a number line plan, I would have stopped my error faster!
Sure A_1 does not include 1, and A_2 does not include 2, and A_3 does not include 3, but A_3 included 2 and 1, and A_22 won't include 9, but every positive integer less than 9.

The Union is Not N :( But The intersection is indeed every real number on the open interval (0,1).

Don't worry. Any help at all is appreciated. Plus, I'm studying this stuff, you're just pulling it from memory. Mistakes are fine ^^;

But yes. A1 is (0,1)
A2 is (0,2) (Aka, it has 1)
Ainfinity... (0,infinity) then?
But what's the difference between that and N?
Ooooohhh... N includes zero...

I see, I see...

Just as a curiosity, (0,infinity) would be the same as N, if we removed 0 from N, yes?

 

[Gib Z]

Perhaps I am using the letter N differently to you lol. N means the set of the Natural numbers, which basically means the positive integers. This set only includes 1,2,3... etc etc

But A_1 includes 0.5, since A_1 means "All real numbers between 0 and 1, not inclusive", which 0.5 certainly satisfies =] But the open interval (0,infinity) has all the Positive integers, and more, like the 0.5 I just said. So that's the difference between (0,infinity) and N =]

(0,infinity) is commonly denoted by \mathrr{R^{+}}, ie the Positive real numbers.

 

[Goldenwind]

Perhaps I am using the letter N differently to you lol. N means the set of the Natural numbers, which basically means the positive integers. This set only includes 1,2,3... etc etc

But A_1 includes 0.5, since A_1 means "All real numbers between 0 and 1, not inclusive", which 0.5 certainly satisfies =] But the open interval (0,infinity) has all the Positive integers, and more, like the 0.5 I just said. So that's the difference between (0,infinity) and N =]

(0,infinity) is commonly denoted by \mathrr{R^{+}}, ie the Positive real numbers.

Your definition conflicted with what my book says (That N = {0, 1, 2, ...}... aka, it includes zero), so I did some research, and apparently we're both right :O
http://en.wikipedia.org/wiki/Natural_number

"In mathematics, a natural number can mean either an element of the set {1, 2, 3, ...} (the positive integers or the counting numbers) or an element of the set {0, 1, 2, 3, ...} (the non-negative integers). The former is generally used in number theory, while the latter is preferred in mathematical logic, set theory, and computer science. See below for a formal definition."

But yeah, I forgot about decimals =/ Wraskley (lol, spelling? XD) little numbers they are...

However, with R+, we then get into the debate about if zero is positive =/
My book says R+ does not include zero... so for my work, I'll be putting down (0,infinity).
However, in all honesty, I don't know what zero is :P

To me, it is either:
- Neutral (Not positive or negative)
- Both (-0 = +0 = 0, therefore it is positive, and negative at the same time)

Ohhhh... *bonks self on head*... but our answer doesn't include zero anyway, so R+ works...

Hah. I love how you say something, and it seems off to me at first, then I try to explain why it seems off, but in explaining, I realize MY OWN mistake, and see how you're right :P

Yay! My thoughts work in proof by contradiction ;)
Yes, I'm a confused child, don't mind me :P

LOL ^^;

 

[Gib Z]

Since you are studying sets here, it is better that you use the "includes zero" definition =]

Your book is very good, R+ does not include zero. It only includes the positive real numbers, of which zero is not. I could add zero as an element to R+, but instead of it being the set of the positive real numbers, it is the set of the non-negative real numbers =] Ahh see how we get around these problems with nice wording!

Ahh doesn't it feel fun to actually grasp a concept and learn something =] These little personal achievements that fill us up with joy =]
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It's a very deep thing, to say you don't know what zero is =] There were books written by Philosophers just asking the question "What is a number?". I can't remember his name, but some mathematician basically excluded mathematicians from that debate by simply saying something like, "Who cares what they are, let's start finding their properties!". =]