Science:Math Exam Resources/Courses/MATH220/December 2009/Question 04 (b)
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Question 04 (b) 

Prove that if A is a denumerable subset of then there exists a set B such that B is a proper subset of A and . 
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 

Recall that a set is denumerable if it is in bijection with the natural numbers. Setting up this bijection should help the proof. 
Hint 2 

Due to this bijection with the natural numbers, we may write

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. Since A is denumerable, there is a bijection with this set and the natural numbers. Under this bijection, we can think of the elements of A as Now let B be the set of elements and create a map from B to A via This map is injective and onto and thus gives a bijection. This means that A and B have the same cardinality completing the proof. 