Does a 5cm straightline have more points than a 3cm straightline?

Question

I recently came across a problem that asked whether, as the title says, a 5cm straight line has more points on it than a 3cm straight line.

(Note: All lines considered here are straight lines even if not explicitly mentioned everywhere)

My way of solving it was to consider the 3cm line superimposed on the 5cm line. Then, every point on the 3cm line would lie on a point on the 5cm line, but there would still be some points on the 5cm line that aren't covered. And this would be true no matter how you superimposed the 3cm line. So, the 5cm line must have more points.

However, another proof I saw made use of constructing a triangle with the 5cm straightline acting as its base. The 3cm line was then drawn parallel to the 5cm base inside the triangle, such that it's each of its endpoints were on the other 2 sides of the triangle. This proof claimed that a line could be drawn to the vertex opposite to the base from every point on the 5cm base and that each of these lines would have to pass through the 3cm line. And so, every point on the 5cm line has a corresponding line on the 3cm line, implying that both lines had same number of points on them.

Now I know that both these answers can't be correct, but I just can't find fault in either proof. So, my question is which of these proofs is correct and most importantly, why?

Thank you!

A proper subset of an infinite set can be in one-to-one correspondence with the whole set, so the first proof doesn't show that the cardinalities of the two sets are different. The second proof is correct. — saulspatz, Jul 30 '21 at 05:13
Thank for such a quick answer @saulspatz. I guess I still have a lot to learn about one-to-one correspondence. — Gaurav Chandan, Jul 30 '21 at 05:28
It's not so much one-to-one correspondence that's the issue, as it is the notion of cardinality of infinite sets. — saulspatz, Jul 30 '21 at 05:32
This might be relevant: $(0,1)$ and $(0,\infty)$ have the same cardinality. In fact, there are just as many points on a $3$cm straight line as there are on the entire $\mathbb R^2$ plane... — ryang, Jul 30 '21 at 05:46

Sayan Dutta · Answer 1 · 2021-07-30T05:39:56.550

5

This is how we deal with infinities. We call two infinite sets to be equal (by equality, here I mean the same cardinality) if there is bijection between them.

Your second line of argument explicitly constructs a bjection between the points on a $5$cm line and a $3$cm line. So, they must be equal.

Your first line of argument only shows that there is an injection from the $3$cm line to the $5$cm line. That doesn't mean there cannot be a bijection from one to the other.

edited Jul 30 '21 at 05:39

answered Jul 30 '21 at 05:29

Sayan Dutta

8,831

2

Rigorously speaking your first sentence is not correct. It is not the definition of two sets to be equal, but the definition of two sets to have the same cardinality (or informally the same number of elements). – Benjamin Jul 30 '21 at 05:36
1

@Benjamin Yes, that's what I meant. I'm placing this edit. – Sayan Dutta Jul 30 '21 at 05:38
This gives quite a bit of clarity. Thanks @SayanDutta. – Gaurav Chandan Jul 30 '21 at 05:41
@GauravChandan You're welcome! – Sayan Dutta Jul 30 '21 at 05:56

score 4 · Accepted Answer · answered Jul 30 '21 at 05:22

Both of your methods are correct, however you draw the wrong conclusion from the first one. Your first proof uniquely takes every point from the 3cm line uniquely to a point in the 5cm line. The conclusion is: The 5cm line has at least as many points as the 3cm line. Now, your second proof uniquely takes every point from the 5cm line to a point on the 3cm line. Hence, similar to before, the conlusion is: The 3cm line has at least as many points as the 5cm line.
Combined, you have proven both lines have the same number of points.
If you look at the second proof closer, the correspondence therein is in fact one-to-one and onto (bijective), i.e., every point is of the 5cm line is mapped to one point of the 3cm line and every point is covered. This directly implies the statement that both lines have the same number of points.

Ah okay, I get it now. Thanks, @Benjamin – Gaurav Chandan Jul 30 '21 at 05:28 — Gaurav Chandan, Jul 30 '21 at 05:28

Does a 5cm straightline have more points than a 3cm straightline?

2 Answers2