# Subset of infinite set

A point. There is an infinite number of lines intersecting it, but this is less than the number of possible lines. How do we represent this in mathematical notation?

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Define $\mathcal{S}_n=\lbrace p|\hbox{ } p\textrm{ is a line in }\mathbb{R}^n\rbrace$ and for $x\in\mathbb{R}^n$, $n\geq2$ define $\mathcal{S}_n^x=\lbrace p|\hbox{ } p\textrm{ is a line in }\mathbb{R}^n\textrm{ passing through }x\rbrace$. Then $\mathcal{S}_n^x$ is a proper subset of $\mathcal{S}_n$, in symbols: $\mathcal{S}_n^x\subset\mathcal{S}_n$ or $\mathcal{S}_n^x\subsetneq\mathcal{S}_n$ (to avoid confusing it with not-necessarily-proper inclusion). Is this what you want to express? – Dejan Govc Jan 31 '12 at 13:21

These are the two meanings I find to this:

1. The set of lines passing through $x$ is a proper subset of the set of lines, similar to how the integers are a subset of the rationals: $$\{L\mid L\text{ is a line passing through } x\}\subsetneq\{L\mid L\text{ is a line}\}$$

2. The cardinality of the set of lines passing through $x$ is less than the cardinality of the set of lines, similar to how the cardinality of the reals is greater than the cardinality of the integers: $$|\{L\mid L\text{ is a line passing through } x\}|<|\{L\mid L\text{ is a line}\}|$$ In some places it is customary to use $\#\mathrm{A}$ or $\mathrm{card}(A)$ to denote the cardinality of the set $A$, in other places it is customary to write, as I did, $|A|$.

Of course you can give names to these sets, and then it can look better. Note that if you talk about lines in $\mathbb R^n$ then the cardinality of the two sets is the same, so the second statement about cardinality is false.

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There's a third meaning, which is is that there is a non-trivial measure defined on the set of all lines, and the measure of the set of lines through a single point is zero. In this sense, the set of all lines through a particular point is "insignificant" relative to all lines. – Thomas Andrews Jan 31 '12 at 14:18
@Thomas: There is a fourth meaning of being a first category set as well; there are many different notions for size. These are the two which I thought are relevant. – Asaf Karagila Jan 31 '12 at 14:22
you could also define a topology on the set of all lines, and then notice that the lines through a particular point are a "nowhere dense set" in the parent set. Or you could define a manifold structure on the set of all lines, and then notice that the set of lines through a point is a manifold of lower dimension. – Thomas Andrews Jan 31 '12 at 14:23
Additionally, you could define a manifold structure on the set of all lines, and note that the set of lines through a particular point is a sub-manifold of smaller dimension. – Thomas Andrews Jan 31 '12 at 14:25
@Thomas: I really don't want to confuse someone that might not know much about mathematics with topology and measure. – Asaf Karagila Jan 31 '12 at 14:25

Infinite cardinal numbers can be tricky. For example, you may say that the set of positive integer has more elements than the set of positive even integers, but the cardinal of $\Bbb{N}$ is the same as the cardinal of $2\Bbb{N}$. Two sets have the same cardinal if there is a bijection between them.

For the sake of simplicity, suppose we are in $\Bbb{R}^2$. In your case, the lines passing through a point can be put in bijection with the angle such line has with respect to $Ox$, for example, and that set is $[0,\pi)$ and its cardinal equals $\aleph=\mathrm{card} \Bbb{R}$.

The set of lines $L$ satisfies $\aleph\leq \mathrm{card} L \leq \mathrm{card}\{(x,y),(z,t) : x,y,z,t \in \Bbb{R}\}=(\mathrm{card} \Bbb{R})^4=\aleph^4=\aleph$. So the set of lines passing through a point, and the set of all lines have the same cardinality.

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