is card(Z) = card(Q)?

Question

is the set of the integers equalt to the set of the rational numbers?
Need a proof!

Is the set of integers equal to the set of rational numbers? I need a mathematical proof! Please don't answer with the trivial answer that the set of integers is smaller! (Besides, I'm not sure that's true)

sorry, I meant the size of the sets, not the sets themselves.

Answer 1

(Edit: Updated, after reading thomasoa's comments. Thanks, I am rusty on this stuff).

Yes they have the same cardinality. To show that two cardnalities are equal, you need to show a bijection. To do so formally, we can show mappings one to one mappings from each set into the other.

To map integers to rationals, just map each integer to itself. (This shows that Card(Z) is less than or equal to Card(Q)).

Then to map rationals one to one to integers. For any positive rational a/b (a,b positive integers), map a/b to (a+b-1)(a+b-2)/2 + a. So 1/1 maps to 1, 1/2 maps to 2, 2/1 maps to 3 and so on.

You can map 0 to 0, and map negative rationals to negative integers in nearly the same way.

A little algebra will show that this is indeed a one to one mapping, and thus Card(Q) is less than or equal to Card(Z).

Thus combining those two facts, you get Card(Q) = Card(Z).

Answer 2

You've asked two seperate questions. The set of integers, obviously, is not equal to the set of rational numbers.

However, the two sets do have the same cardinality - they are the same "size."

You can come up with a simple 1-1 map from Z->Q, namely, inclusion. So all you need is a 1-1 map from Q->Z to prove the two are the same.

Given a rational number (+/-)(p/q), where p and q are relatively prime positive integers, you send this number to (+/-) 2^(p-1) * 3^(q-1), where the sign equals the sign of the original rational number. Send the rational zero to the integer zero. This is clearly 1-1 so you are done.

1/1 -> 1
2/1 -> 2
1/2 -> 3
3/1 -> 4
1/3 -> 9
3/2 -> 12
2/3 -> 18
...

Answer 3

Strangely enough, the cardinality of the set of integers is equal to the cardinality of the set of rational numbers. Both are aleph-null, that is, countable infinity.
http://pirate.shu.edu/projects/reals/infinity/countble.html
http://www.maa.org/devlin/devlin_01_06.html
The cardinality of the set of irrational numbers is higher, aleph-one.
Of course, the sets themselves are not equal, since in order for sets to be equal, each must be a subset of the other. Z is a subset of Q but not vice versal

Answer 4

Mathematically, I suppose you could say the cardinals are equal, since Card(Z) = Infinity and Card(Q) = Infinity.

There are infinite integers and infinite rational numbers. You could say that they are equal if you take them in this way.

Answer 5

no, it is proper said as

The set of integers is a subset of the set of rational numbers.

Answer 6

a rational number does not necessarily be an interger.
3/2 is a rational number, but it's not an interger.

so the set of raional numbers is bigger than the set of intergers.