It might be a silly question but I was just wondering if this was unprovable or false...

Thanks guys

Printable View

- Dec 31st 2006, 04:55 AManthmooAre half of all natural numbers even?
It might be a silly question but I was just wondering if this was unprovable or false...

Thanks guys - Dec 31st 2006, 05:17 AMCaptainBlack
- Dec 31st 2006, 05:56 AMTD!
The question makes about as much sense as "Is half of the integers a natural number?"

- Dec 31st 2006, 07:33 AMThePerfectHacker
- Dec 31st 2006, 07:39 AManthmoo
- Dec 31st 2006, 07:41 AManthmoo
- Dec 31st 2006, 07:46 AMThePerfectHacker
I am not sure what you are asking.

There are several sizes of infinite sets. For example, the reals numbers (rationals and irrationals) are infinite. And the integers are infinite. But the reals are larger in size then the integers. Yes, even though they are infinite sets they have different sizes. The smallest type of infinite is represented by $\displaystyle \aleph_0$ and they are called countable. Everything in the universe is countable. If the stars are infinite then they are countable, meaning you can assign a number to each one. Since both the integers and even integers can be listed (counted) they are countable and hence are the same in size. - Dec 31st 2006, 08:45 AMTD!
Perhaps ThePerfectHacker's explanation sounds a bit strange to anthmoo, let me try to clarify this new concept a bit.

Given a certain set (for example, the set of integers, reals, the set {1,2,3}, the empty set, ...) you'd like to be able to know its size, more generally put: you want to quantify how "large" it is.

For finite sets, this is easy: you can explicitly state its size, as the number of elements it has. The set {0,1,2,3,4,5,6,7,8,9} has size 10 while the set {1,2,3} has size 3, since they respectively have 10 and 3 elements. This easy way of quantifying a set is lost when dealing with infinite sets, since both the naturals (with or without zero, e.g. {0,1,2,3,...}) and the integers (namely {...,-2,-1,0,1,2,...}) have an*infinite*number of elements.

Mathematicians (Cantor did a lot on this field) have worked out a concept which allows to quantify "infinity" as well, i.e. to be able to say that certain infinities are 'larger' than others. This concept is called**cardinality**(google it for more info). It is important to note that the true notion of "size" in its classical sense is lost, which is why you'll never hear me say that the integers and the naturals "are of the same size". Although this is mathematically correct if you're referring to their cardinality, many people confuse this wording with the classical size and then it seems unlogical to them.

To come back to ThePerfectHacker's explanation, we first introduce the "smallest" possible form of infinity, this is the one of the natural numbers. We call this type of infinity**countable**, exactly because you can 'count' the natural numbers (1,2,3,...). Now we introduce an important definition: we say that another set X has the same cardinality as the naturals, if you can find a bijection between the sets (note that their exist subtle differences in the definitions, whether you allow finite sets as well - you could then use an injection). That is, you can put the elements of both sets into a one-to-one correspondence.

Example: the cardinality of the integers is the same as that of the naturals (in more popular wording: their "size" is the same). Indeed, I can construct the following bijection (N<->Z): 0<->0, 1<->1, 2<->-1, 3<->2, 4<->-2, ... This way, every natural number has an integer partner and vice versa, which we wanted. A bit more subtle is the following result: also the rationals (Q) has the same cardinality as N (and Z), thus is countable.

Next step: the real numbers (R). It can be shows (search for Cantor's diagonal argument) that you cannot find a bijection between the naturals and the reals. This leads to the conlusion that the reals form a new, bigger kind of infinity. We call this**uncountable**. An important question used to be: is there an infinity strictly*between*the one of N and R, or is the cardinality of R the first larger kind of infinity. This is known as the**continuums hypothesis**, which appeared to be independent of the current axioms, it's neither 'correct' nor 'false'.

Now, back to your question: what does this have to do with even and odd numbers? Well: you can easily show that the cardinality of the set of even numbers (let me denote E = {0,2,4,6,...}) is the same as the one of the naturals, and of the odss of course (O = {1,3,5,...}). I'll show the bijection E<->N: e = 2n. Indeed: 0=2*0, 2=2*1, 4=2*2, ... Thus: you can assign an even number to every natural number and vice versa.

Conclusion: although we intuitively feel that half of the natural numbers is even, half is odd (so that indeed the sum if the complete set of naturals), isn't very "rigorous". To treat sizes of infinite sets, we rely on cardinality and as you have seen, that shows - popularily said - that there are as many even, as odd, as natural numbers. I repeat: saying it like this can make it sounds unlogical, which is why it's safer to just talk about cardinality, not size.

Hope this helps :) - Dec 31st 2006, 09:33 AMCaptainBlack
There is a one-to-one mapping between the even integers and the integers

as a whole, so in that sense they are equi-numerous, and the fraction that

are even has no meaning.

Despite what others have posted I would go with the explanation in my

previous post (the limit interpretation is one in which the statement

makes any sense).

RonL - Dec 31st 2006, 09:50 AMThePerfectHacker
- Dec 31st 2006, 09:55 AMThePerfectHacker
To add to TD! post....

There are some "sets" that are so large that cannot be contained in other sets. So large that you cannot even give them a number (in this case called "cardinal" number, as TD! was saying). For example, the set of all sets is TOO large and is not contained in other sets. In this case these are called "categories". - Dec 31st 2006, 09:55 AMCaptainBlack
- Dec 31st 2006, 10:03 AMThePerfectHacker
You think I did not see the changed post?

----

A function $\displaystyle \phi: X\to Y$ between two non-empty sets is:

"One-to-one" when it is an**injective***) function. Meaning $\displaystyle \phi(x)=\phi(y)\to x=y$.

"Onto" means**surjective***) functuin.

Meaning $\displaystyle \forall y\in Y \exists x\in X , \phi(x)=y$.

"One-to-one and onto" is a combination of both and means**bijective***) function. This is also called**one-to-one correspondce**. Because when we have a bijective we can pair elements together (hence the name).

*)These terms were first used by Bourbaki. - Dec 31st 2006, 10:08 AMCaptainBlack
And you miss the implication of my wording which was not fully formal,

but it was that there is a one to one mapping**between**the even naturals

and the naturals, not from the evens to the naturals. Next time I may be

more explicit.

OK ok ... I made a slip:p

RonL