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I'm on a conversion MSc and haven't done math in a while, so apologies if this is an easy Q but I'm stumped. I've literally only just learned what radians and sets are- I think I've done a decent job of getting the basics of each, but I'm struggling with applying them to this problem.

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Set $A, B$ are fine, but I'm pretty stumped by set $C$. I suppose the relationship $1 / 2\pi$ isn't rational, but I'm lost as to how that helps me figure out the cardinality of the set. I'm guessing that as the relationship between how many points can be drawn on the circumference isn't rational that giving the cardinality isn't as simple as giving a number?

I've got as far as sort of testing a few ideas:

  • there aren't an integer number of radians in the full circle, so how can I have a finite series of points to give the cardinality?

I don't really want the answer as I'm trying to figure it out for myself, but any hints constructive to my learning would be appreciated!

Shubham Johri
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Jack Brown
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    I'll try to be very vague to not spoil it: There is not an integer multiple of 240 degrees in a full circle, but if that was the angle then your cardinality would be 3. – preferred_anon Oct 22 '20 at 19:30
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    Your set has the points $(\cos 1,\sin 1), (\cos 2,\sin 2), (\cos 3,\sin 3),\dots$ Does your set ever repeat? Is it possible that $(\cos n,\sin n) = (\cos m, \sin n)$ for two different positive integers $n,m$? What would the difference between $n$ and $m$ have to be for that to be true? – JMoravitz Oct 22 '20 at 19:31
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    There are $2\pi$ radians in a full rotation. Suppose that you have two points $i_1w,i_2w$ with $i_1\ne i_2$. These two points are coincident iff $i_1w-i_2w$ is a multiple of $2\pi$ (angle traversed in one rotation). Is it possible? – Shubham Johri Oct 22 '20 at 19:32
  • @ShubhamJohri Thanks for the hint! I can see how it's taken me closer to the answer. Would I be correct in saying no, no two integer multiples of a radian could ever be exactly a multiple of 2$\pi$ away from each other? My intuition is that the cardinality would keep growing as the set never repeats. – Jack Brown Oct 28 '20 at 15:31
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    Your intuition is absolutely correct! – Shubham Johri Oct 28 '20 at 19:24
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    @JackBrown I have put my comment as an answer. If it is satisfactory, consider accepting it by clicking the tick-mark button next to it. – Shubham Johri Oct 28 '20 at 22:16
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    @ShubhamJohri Woop that's a good feeling! I have done so, and thanks for being patient with a newcomer. – Jack Brown Oct 29 '20 at 14:23
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    You are welcome. – Shubham Johri Oct 29 '20 at 17:25

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Putting my comment as a hint in an answer.

There are $2\pi$ radians in a full rotation. Suppose that you have two points $i_1w,i_2w$ with $i_1\ne i_2$. These two points are coincident iff $i_1w−i_2w$ is a multiple of $2\pi$ (angle traversed in one rotation). Is it possible?

Shubham Johri
  • 17,659