Help Me Understand?

For the discussion of math. Duh.

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snow5379
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Help Me Understand?

Postby snow5379 » Tue Nov 15, 2011 4:17 pm UTC

I'm not sure I agree with this:

If for every unique apple there's a unique orange then there are as many or more oranges as apples.


It's works for integers I guess... but does that mean it automatically should work for everything else? How would you prove something like this? I'm not very smart and I'm just trying to understand... so any help would be welcome.

skullturf
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Re: Help Me Understand?

Postby skullturf » Tue Nov 15, 2011 4:44 pm UTC

snow5379 wrote:... but does that mean it automatically should work for everything else?


That's a fair question.

It's not so much "this works for integers, so we have to assume it works for infinite numbers or sets, too."

It's more a case of: After experimenting with different notions of what it means for one set to be larger than or equal to another, we have found that a particular notion works well for infinite sets, so we decide to use it.

(There wasn't an enormous amount of context, but I took the post as referring to notions of "greater", "equal", or "less" for infinite sets.)

mfb
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Re: Help Me Understand?

Postby mfb » Tue Nov 15, 2011 8:05 pm UTC

Write a unique integer on each apple and each orange.
Yes, it works - as long as you can assign a unique number to each apple and orange (note that this is true for apples and oranges with finite, distinct volume in a finite observable universe), otherwise it cannot be transferred.

But as skullturf already pointed out, the definition for "as many or more" is the more general definition here. You can use it for apples and oranges even if it is not possible to write a unique integer on each.

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314man
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Re: Help Me Understand?

Postby 314man » Wed Nov 16, 2011 6:44 am UTC

Isn't this a counter example?:

Say the numbers on apples {1, 1, 2}, and on oranges {1, 2} (each number representing the number written on the fruit). There are two unique apples and two unique oranges, but there are more apples than oranges

Yesila
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Re: Help Me Understand?

Postby Yesila » Wed Nov 16, 2011 7:03 am UTC

314man wrote:Isn't this a counter example?:

Say the numbers on apples {1, 1, 2}, and on oranges {1, 2} (each number representing the number written on the fruit). There are two unique apples and two unique oranges, but there are more apples than oranges


I think you actually have three unique apples there. 2 of them just happen to have the same number on them. For a more concrete example of this in action think about people and names. I happen to know at least 3 people with the name "Mike" but even though they each have the same name... they are all unique people.

Yesila
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Re: Help Me Understand?

Postby Yesila » Wed Nov 16, 2011 7:15 am UTC

snow5379 wrote:I'm not sure I agree with this:

If for every unique apple there's a unique orange then there are as many or more oranges as apples.


It's works for integers I guess... but does that mean it automatically should work for everything else? How would you prove something like this? I'm not very smart and I'm just trying to understand... so any help would be welcome.


To prove something like this translate the abstract words into actual mathematical constructs. In this case "for every unique apple there's an orange" is telling us we have a function from the space of apples to the space of oranges. Since there happens to be a unique orange we can further say that this function is injective. If we remove from the space of oranges any oranges that are not in the range of this function then the function becomes bijective, and we know the space of apples and the space of oranges (with possibly some removed) have the same cardinality. Adding back in any removed oranges (may) increase the number of oranges but it certainly won't decrease the number of oranges so the conclusion follows.

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Xanthir
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Re: Help Me Understand?

Postby Xanthir » Wed Nov 16, 2011 4:39 pm UTC

314man wrote:Isn't this a counter example?:

Say the numbers on apples {1, 1, 2}, and on oranges {1, 2} (each number representing the number written on the fruit). There are two unique apples and two unique oranges, but there are more apples than oranges

If you're responding to mfb, then no, that's not a counter-example, because mfb specifically said "write a unique integer on each apple and each orange". You did not do this.
(defun fibs (n &optional (a 1) (b 1)) (take n (unfold '+ a b)))


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