cardinality (complicated question(s), cute answ...

I was going to say no to the first, but seeing the injection made changed my answer.

However, I think the answer to #2 is No, since any function capable of creating any rational number from naturals has to include two independent variables, and it seems likes injections are limited to ONE variable in a one-to-one correspondance. In intecting rational numbers to natural numbers, you'll find any system will have multiple rational numbers for the same natural value.

The third is definitely no, but I couldn't tell you exactly why in precise mathematical language, but the gist is that the set of real numbers is infinite on a larger scale than any rational set. (The mathematicians refer to a series of alephs, or degrees of infinity at this point.)

And while I was good at math, the fourth question I fail to grasp at the moment. Been too long, and it's a bit too scholarly for my comprehension.

Originally posted by geepamoogle
I was going to say no to the first, but seeing the injection made changed my answer.

However, I think the answer to #2 is No, since any function capable of creating any rational number from naturals has to include two independent variables, and it seems likes injections are limited to ONE variable in a one-to-one correspondance. In intecting rational ...[text shortened]... il to grasp at the moment. Been too long, and it's a bit too scholarly for my comprehension.

You are correct about the first, but the answer to the second question is also yes. It's a tricky bijection if you write it out, but there is a more elegant solution:

N={0,1,2,3,4,5,...}
Z={..., -3, -2, -1, 0, 1, 2, 3,...}
Q={rationals}

It's easy to see that N ~ NxN (there is a bijection between N and pairs of elements in N. Think about it, this is the crucial step).

From above we know then that NxN ~ ZxN (~ ZxZ)

Now all rational numbers can be expressed as the divisoin of an element in Z by an element in N, in other words it is represented by a pair of numbers, an element from ZxN.

This shows that there certainly are enough elements in N to get to all elements in Q.

The problem in R={reals} is that, indeed, it is too big.
We'll discard most of R and show that that collection of numbers is still too big. Take all reals of the form 0,... with only 0's and 1's as decimals.

Suppose we can make a bijection, that would mean we can make a complete row of all reals of above form.

Make a new element A = 0,abcd... by looking at the elements in the row. Make a different from the first decimal of the first element in the row. Make b different from the second decimal of the second element in the row. And so on.

This way A will be different from all numbers in the row. This means that A isn't in the row. This is a contradiction with the assumption that we could place all elements in a row.

Thus, such a row does not exist, and there is no bijection from N to R.

(Apologies for any errors, it's late here)

Hmmm, thought about it some, and I may have one which maps 0 to 0, positive to positive and negative to negative.

The negative mapping will mirror positive save for the sign change, and 0 0 is straightforward.

So from 1 on is as follows..

1/1, 1/2, 2/1, 1/3, 3/1, 2/3, 3/2, 1/4, 4/1, 3/4, 4/3, 1/5, 5/1, etc..

The rules of the order are as follows are as follows:

1) Natural 1 is assigned rational 1 for the bijection.
2) First select your larger natural, starting from 2 and going up by 1 after you've exhausted all possibilities.
3) Select your second natural number from 1 to the larger, omitting any numbers which aren't relatively prime (GCF = 1).
4) Assign the ratio of the smaller to the larger to the next number.
5) Assign the inverse to the next number.

There there you have a one-to-one Natural-to-Rational function. You're right, that is counter intuitive..

Originally posted by geepamoogle
Hmmm, thought about it some, and I may have one which maps 0 to 0, positive to positive and negative to negative.

The negative mapping will mirror positive save for the sign change, and 0 0 is straightforward.

So from 1 on is as follows..

1/1, 1/2, 2/1, 1/3, 3/1, 2/3, 3/2, 1/4, 4/1, 3/4, 4/3, 1/5, 5/1, etc..

The rules of the order are as fol ...[text shortened]... ou have a one-to-one Natural-to-Rational function. You're right, that is counter intuitive..

Nice work 🙂

The one i always use doesn't do the negatives-to-negatives thing 🙂