Question

IIs the infinite set of whole numbers (0,1,2,3,4,5,6) smaller than, larger than or the same size as the infinite set of integers {0,1,-1,2,3,-3,4,-4}

Answer 1

infinite set of whole numbers (0,1,2,3,4,5,6) --> not written as an infinite set.

(0,1,2,3,4,5,6) is a finite set consisting of the whole numbers 0,1,2,3,4,5, and 6.

Check the original question.

Answer 2

> {0,1,-1,2,3,-3,4,-4} is also written in the form of a finite set.

Answer 3

@Directrix Despite his mistake, I think the question is still about whether the infinite set of whole numbers is larger/equal in size in comparison to the infinite set of integers. Would you agree?

Answer 4

I would like to know if he is studying Cantor and transfinite numbers.

Answer 5

Going by this assumption, we can say that the cardinality of both the infinite set of integers and the infinite set of whole numbers is the same because there exists a bijection such that \(\bf f:\mathbb{Z} \rightarrow \mathbb{N} \).

Note: I'm using the set of Natural numbers for the set of whole numbers and I'm including the zero since the set that you typed up also includes the zero. Despite there is no set conclusion as to whether the set of natural numbers includes zero or not, we will assume that it does.

Answer 6

@JonesR Read the first paragraph at this link:

http://www.britannica.com/EBchecked/topic/93251/Georg-Cantor/1090/Transfinite-numbers

Answer 7

wikipedia is pretty good too lol =p @Directrix

Answer 8

> wikipedia is pretty good

@genius12 Not for me. Wikipedia is not an *academic* research site.

And, zero is *not* a natural number.

Answer 9

@Directrix ik what you mean lol...and my teachers use it =.=

Answer 10

@genius12 Then, you should alter some of the Wikipedia items your teachers reference and then ask them about it.

Answer 11

@Directrix When I say "...and my teachers use it", I mean that they tell me to not use it since it is not a scholarly source but at the same time, they use it to get a quick answer.

Also, like I already mentioned, it's debatable whether the zero is included or not but there is no conclusion as to whether it is. Some use the set of natural numbers as the set of whole numbers, i.e. include the zero, while others use it as the set of counting numbers which excludes 0.

Answer 12

its not debatable, Natural is a definition not a theorem

Answer 13

hence having a definition of whole numbers:)

Answer 14

well his point is that it's mainly convention as to whether \(0\in\mathbb{N}\) and that the debate is whether it *should* be included or not :-p

Answer 15

yep

Answer 16

My understanding: If you add the whole set of integers you get 0 as each number has a negative. Therefore the set of whole numbers would be larger. Simple answer :).

If you mean by number of elements the set of Integers would be larger as it has the negatives as well.

Answer 17

interesting approach... I wonder if much work has been done in considering cumulative sums as a measure of the 'size' of a group... :-p anyways in reality they have *the same number of elements* since you can create a one-to-one mapping (i.e. a bijection) between the sets. consider, for example,:$$f(n)=\begin{cases}n/2\qquad\qquad\quad\ \text{if }n\text{ is even}\\-(n-1)/2\qquad\text{if }n\text{ is odd}\end{cases}$$this generates all of the integers without repetition despite only requiring all the natural numbers hence the two sets of numbers must have the same cardinality