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Find the cardinality of the set of all straight lines in $\mathbb R^2$.

Here's what I did:

Let $M$ be the given set.

$$M \sim\{y=ax+b, \ a,b\in \mathbb R \}\cup\{x=c, \ c\in\mathbb R \}$$

So:

$$|M|=|\{(a,b) \ a,b\in \mathbb R\}|+|\{c, \ c\in\mathbb R\}| = \frak c\cdot\frak c +\frak c =\frak c$$

Thomas Russell
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GinKin
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1 Answers1

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Another approach is this. Let $S$ be the set of all lines. Then one can construct a surjection $[0,2\pi)\times \Bbb R^2\to S$, as follows: pick a point in $\Bbb R^2$; and an angle, and obtain a line. Of course any line can obtained in this way. (Note this is not an injection!). But we also have an injection $\Bbb R\to S$ by taking a real number $r$ to the horizontal line $y=r$. Thus, we have $$\mathfrak c=|\Bbb R|\leqslant |S|\leqslant |[0,2\pi)\times \Bbb R^2|=\mathfrak c^3=\mathfrak c$$

Pedro
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    Some of these lines are the same, so the conclusion of your argument should be ".. there are at most $\mathfrak c$ lines." Since there are also at least $\mathfrak c$ lines, such a all the horizontal ones, there are exactly $\mathfrak c$ in total. – hmakholm left over Monica Mar 11 '14 at 18:20
  • @HenningMakholm I guess it is better now. – Pedro Mar 11 '14 at 18:25