One-to-One Mapping

Question

I have a conceptual kind of a question in one of my lectures, where you have two line segments one is $(0,1)$ and the other $(0,2)$ which of course includes $1$ in the segment. It then ask what the number of points are in each segment, which I assume is infinite and thus the number of points in both is the same.

However, I'm not sure if that's true or not - since it seems to be a trick question, especially since it's followed by asking if the segment which has more points $(0,1)$ or $(0, \infty)$. Any help would be appreciated!

Thanks,

Answer 1

Any two line segments have the same number of points: that is, the points in one segment can be put in one-to-one correspondence with the points in the other segment. (You'll soon learn that "infinity" is not a good answer for "what is the number of points?" as there are many different "levels" of infinite sizes).

For the line segments $(a,b)$ and $(c,d)$, take the function $$f(x) = c + \left(\frac{x-a}{b-a}\right)(d-c).$$ Verify that this is one-to-one and onto.

Can you think of a continuous function that maps $(0,1)$ onto $(0,\infty)$ one-to-one? $f(x) = \frac{1}{x}$ doesn't quite work, but maybe something similar?

Answer 2

The fact that $(0,1)$ has infinitely many points is correct of course. However infinity is a tricky concept.

1. In finite sets, if two sets have $k$ elements then any function is injective (one-to-one) if and only if it is surjective (onto) if and only if it is a bijection (injective and surjective). This means that two sets are of the same size if and only if you can write them each in one column of a table, without repetitions and the columns will be of the same length.

2. The last point brings us to infinite sets, where one can cook all sort of crazy mappings. We say that two sets have the same number of elements if and only if there is a bijection between them. That is, we can write them in infinite columns without repetitions and have the two columns in the same length.

3. There are different sizes of infinity, in fact there are infinitely many of them. Under some common assertions $|\mathbb N|$ is the least infinity, denoted by $\aleph_0$.

4. $\mathbb N$ has the same number of points as $\mathbb N\setminus\{0,1,2\}$, as well $\mathbb Z$ and even $\mathbb Q$.

5. $|\mathbb R| = |(0,1)| > |\mathbb N|$. This is an important fact in mathematics. No matter how you write to write them in columns, the one in which $\mathbb R$ is vastly longer than the column of $\mathbb N$.