Which is there more of?

Question

odd numbers, even numbers or all the whole numbers?

Answer 1

The set of odd numbers, even numbers and whole numbers all have the same cardinality (that is, there is a bijection between them)

There is a bijection between the whole numbers and the odd numbers as follows:

f(t)=2(t)+1; t an integer
This is injective since suppose f(t)=f(s).
Then 2(t)+1=2(s)+1
=>t=s
Indeed, it is surjective, since any odd number can be written in the form 2(m)+1, where m is an integer (whole number), and thus f(m)=2(m)+1
(a bijection is a function that is both injective and surjective, that is, it is 1-1 and onto)

Similarly with the even numbers:
Define g(t)=2t; t an integer
Then this is injective, for suppose g(t)=g(s)
Then we have 2t=2s=>t=s
Indeed, it's surjective because any even number can be written as 2m, where m is an integer.
Thus, for any 2m an even number, 2m=g(m).

In fact, each of these infinite sets are denumerable, that is, there is a bijection from the natural numbers into the set of the odd numbers, there is a bijection from the natural numbers into the set of even numbers and there is a bijection from the natural numbers into the integers.

So you could say that the infinite sets of odd numbers, even numbers and integers all contained the same "number" of elements, even though this "number" is actually an infinite number.

Answer 2

You have to think first what "more of" means in the discussion of infinite sets. The way mathematicians deal with this is they say a set a is >= another set b if the members of b can be put in a 1-1 correspondence with members of a subset of a. So, for instance, the set {4, 5, 6} is <= the set {1, 2, 3, 4, 5} because you can make a correspondence 4 => 1, 5 => 2, 6 => 3 of {4, 5, 6} with the subset {1, 2, 3}. For finite sets you can see they have the same numbers of elements if a <= b and b <= a, which is just what you expect.

Using the same rules for infinite sets, you can make a corresponcence of odd numbers with even numbers 2n-1 => 2n, n >= 1, so the odd numbers are <= even numbers. Similarly, 2n->2n - 1 maps every even number uniquely to an odd number, so evens <= odds, and by the above rules they have the same number of elements.

Odd numbers can be mapped to a subset of the whole numbers by 2n - 1 => 2n - 1, and the whole numbers can be mapped to a subset of the odds by n => 2n + 1, so the whole numbers and the odds have the same order. Similar reasoning shows lots of other sets, such as the squares and the powers of 2, among others also have the same number.

The next set of numbers strictly larger than the integers is the reals. There is no map of real numbers into a subset of the whole numbers.

Answer 3

The same number of each!
Since each set (odd nos, even nos, all whole nos) is infinite in size, we need a new definition of 'how many'.
We use 'one-one correspondence'. For example, the set {cat, dog, eagle} has three members because its elements can be put in 1-1 correspondence with the set {1,2,3}:

cat <===> 1, dog <===> 2, eagle <===> 3

Now look at odd numbers, {1,3,5,7 ...} and set up a one-one correspondence as follows:
1 <===> 1, 3 <===> 2, 5 <===>3, 7 <===>4, ...etc.

You can work out for yourself how even numbers do the trick.

Learn more by finding out about Georg Cantor and transfinite sets.

Answer 4

Whole number (if within 2 intervals)

as whole number comprises of odd numbers and even numbers

For example take the intervale between 1 and 20 inclusive

The number of whole numbers = 20
The number of odd numbers = 10
The number of even numbers = 10

Theoretically the is infinite numbers of each but within a certain interval there is always more whole numbers

Answer 5

Whole number. Since, the set of whole numbers include all natural numbers including zero.

Answer 6

whole numbers of course because you add odd and even numbers to get whole numbers.

Answer 7

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Answer 8

There is an infinite number of all of those, so there is no more or less of any of them I think!

Answer 9

all are equal in quantity, and the total numbers are infinite

Answer 10

can you wait 10 years for this answer, I have to count them to find out!