Science:Math Exam Resources/Courses/MATH220/December 2010/Question 05 (a)
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Question 05 (a) 

Prove that the set is countable. 
Make sure you understand the problem fully: What is the question asking you to do? Are there specific conditions or constraints that you should take note of? How will you know if your answer is correct from your work only? Can you rephrase the question in your own words in a way that makes sense to you? 
If you are stuck, check the hints below. Read the first one and consider it for a while. Does it give you a new idea on how to approach the problem? If so, try it! If after a while you are still stuck, go for the next hint. 
Hint 1 

A set is countable if it can be written in bijection with a subset of the natural numbers, or any other countable set. Can you rewrite the given set and set up a bijection with an already known countable set? 
Hint 2 

Recall that the integers is a countable set. Setting up a bijection between S and the set of integers is your best bet. 
Checking a solution serves two purposes: helping you if, after having used all the hints, you still are stuck on the problem; or if you have solved the problem and would like to check your work.

Solution 

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Please rate my easiness! It's quick and helps everyone guide their studies. Firstly, we examine the set
We claim that where . The inclusion that is easy from trigonometry, we know that . Conversely, notice that is a periodic function. So solving requires only that we solve this between 0 and and then use periodicity. We know that on the interval , we get only two solution at and so taking multiples of this, we have that for an integer give all the solutions. This is precisely the elements of and thus we have shown that equals . Now the mapping from to is trivial. We can map
and in the opposite direction,
These maps are injective, surjective and thus are bijective. Thus, is countable since it is in bijection with a countable set, namely . Aside: A quick example showing that injects into is given by , if and if . That is, maps to and maps to . 