Is the set of points or the set of lines on a plane "larger",or there is a 1-1 correspondence between lines and points?
[Math] Is the set of points or the set of lines on a plane “larger”
elementary-set-theory
Related Solutions
In mathematics, we normally describe a line as, for example, the set of points $\{(x,2x+1):x\in\mathbb R\}$:
Then the function $\phi:x\mapsto(x,2x+1)$ is an easy bijection between the line and $\mathbb R$.
But what if we defined it as the set of points $\{(x,2x+1):\mathbb Q\}$ instead? Then it would be in bijection with $\mathbb Q$ instead. So what would it look like then?
Oh. It's the same. The rational numbers form a dense set (for any rational numbers $x,y$ there is a rational number between them (e.g., $\frac{x+y}2$), so they do 'fill in' space in some way. But we would then be introducing more problems, such as:
The function $x^2-2$ would have no zeroes. That certainly doesn't coincide with our intuition from looking at the graph:
In addition, the function defined by taking
$$ f(x)=\begin{cases}0&x^2<2\\1&x^2>2\end{cases} $$
would be continuous! In order to get round these, we introduce the real numbers, which are complete in the sense that any Cauchy sequence - i.e., any sequence that you would intuitively expect to converge to a value - does in fact converge.
Note that we could have used the algebraic numbers or some other countable set instead of the rational numbers, but would have run into similar problems. We could even use the definable numbers, which are a countable set: in that case, we wouldn't be able to construct these sorts of counterexamples to our intuition, as we could turn them into definitions of undefinable numbers; however, the definition of the real numbers isn't that hard, and it's what mathematicians are used to, and it allows us to prove lots of beautiful results, so that's what we use.
For a discussion on the use of real numbers in the form of a dialogue, see here. I'll just give the closing line.
Mathematician: So, finally, we arrive at the following justification for real numbers. 1. We must go further than just the rationals. 2. When we do so we introduce certain procedures that give us new numbers. 3. Formalizing these, we end up with the monotone-sequences axiom, or something equivalent to it. 4. This axiom is not as precise as it seems, since the notion of an arbitrary monotone sequence, even of rationals, is not precise. 5. There is no need to make it precise, because we know how to reason in terms of arbitrary sequences. 6. That allows us to define the real numbers we have a use for, even if it gives us a lot of junk as well. 7. In fact, we don't really know what junk it does give us, and it's not even clear that it makes sense to ask.
You can List all lines in the plane this way:
Use the fact that a line is described uniquely once you know its slope, and one of its intercepts with the axes. The slope is indexed by the Reals, and so is the intercept, say the x-intercept. So we count all the possible pairs ( slope, x-intercept), and show it is equal to $|\mathbb R|$
1) Consider all lines thru the origin $(0,0)$. These are described uniquely by their slope, and there are $|\mathbb R|$ of them, since the slope is parametrized by the Reals.
2) From 1) , we can cover all other cases of lines not going thru $(0,0)$ , by considering all possible ( say x-) intercepts of a line thru any point, with fixed slope$m$. For every line in 1), there are $\mathbb R$ lines not going thru the origin, but with the same slope. This means there are $|\mathbb R|\times |\mathbb R|=|\mathbb R|$ total lines in the plane.
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Best Answer
Let's start building up a way of describing the set of all lines in the plane. There are two types of lines: vertical lines and non-vertical lines.
The set of all lines in the plane is $S \cup T$, and the set of all points in the plane is naturally identified with $\mathbb R \times \mathbb R$. Now, what do you know about a union of two infinite sets? What do you know about the Cartesian product of two infinite sets? With that information you should be able to put the pieces together and answer the question.