2007-09-10 03:25:17 · 9 answers · asked by Dr D 7 in Science & Mathematics ➔ Mathematics
No, and in fact there are as many even numbers as there are integers.
Whilst this is immediately counter-intuitive, you have to remember that these are infinite sets, and infinity is a very strange thing.
The size of an infinite set is called its cardinality, and all countably finite sets are said to have cardinality aleph, the hebrew letter 'a'. The first countably finite set is the natural numbers-1,2,3,4 and so on. To prove that the set of even numbers has the same cardinality as the natural numbers, you have be able to set up a bijective function between the two sets. (As is required of any two sets to be equal in size). For the case of the even numbers, this is simply N->2N, or simply double each number, i.e:
N : 1 2 3 4 5 6 ...
2N: 2 4 6 8 10 12 ...
Now a bijection can be set up between the set of even numbers (which I will call E) and odd numbers, namely E->E-1, taking one from each element:
E : 2 4 6 8 10 12 ...
E-1: 1 3 5 7 9 11...
And so the set of even numbers and odd numbers are equal in size not only to each other, but to the set of natural numbers as well (also to the set of integers)
Should want to know more about this sort of thing, try the Continuum Hypothesis. Hope I've helped.
2007-09-10 04:03:53 · answer #1 · answered by Anonymous · 3⤊ 0⤋
For counting value N, there is exactly one even element 2N and one odd element 2N+1. For each of the two sets, composed of even and odd elements respectively, N does not terminate since it can continue ad infinitum. At each step (in N), the number of elements of each set is obviously equal. The real question here is, which I assume you are REALLY asking, is what happens AT infinity? Not an easy question to tackle. Based on intuition and the concept of limits, I would say no, the sets have an equal number of elements. But no numerical value can be ascribed to infinity, so the concept of equality fizzles away at infinity, as I'm sure you well know. However, infinity being the strange 'quantity' that it is, I would bet (and challenge, of course) anyone that they could prove (at least loosely) that the answer to this question is ambiguous: both yes and no.
2007-09-10 11:17:12 · answer #2 · answered by Not Eddie Money 3 · 3⤊ 0⤋
Dr D, you know very well that since there is a one-to-one correspondence between the set of even integers and the set of odd integers, the two sets have the same cardinality. So, what are you really looking for?
2007-09-10 10:45:11 · answer #3 · answered by Tony 7 · 2⤊ 0⤋
Since there is an infinite number of both even and odd INTEGERS (which is what I assume you mean by 'numbers'), then there are an equal number of each.
Think of it this way: For EVERY even number, there is exactly one odd number which is equal to the even number + 1, and vice-versa
2007-09-10 10:36:22 · answer #4 · answered by tinfoil666 3 · 2⤊ 1⤋
No, there is a one to one correspondence between the set of even numbers and odd. In fact, there is a one to one correspondence between the set of even numbers and the set of integers.
2007-09-10 11:13:17 · answer #5 · answered by swd 6 · 3⤊ 0⤋
The cardinality of both even and odd numbers is the same.
Thus, the ans to the above question is "no".
2007-09-10 10:34:32 · answer #6 · answered by MH 1 · 5⤊ 1⤋
from what number to what number? If you're pertaining to all numbers.. then no
2007-09-10 10:48:13 · answer #7 · answered by Grand Phuba 5 · 1⤊ 0⤋
the number og them is undefined, coz it's infinite ... is infinite larger than infinite ?? ... what do you think ?
2007-09-10 10:51:58 · answer #8 · answered by Alicia 3 · 1⤊ 1⤋
imposible to tell dr d
2007-09-10 10:32:18 · answer #9 · answered by Steelbro 2 · 0⤊ 4⤋