MATH220 December 2011
• Q1 (a) • Q1 (b) • Q1 (c) • Q1 (d) • Q1 (e) • Q1 (f) • Q2 (a) • Q2 (b) • Q3 • Q4 • Q5 (a) • Q5 (b) • Q6 (a) • Q6 (b) • Q7 (a) • Q7 (b) • Q8 •
Question 05 (a)

State what is means for a set A to be denumerable.

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 hint below. Consider it for a while. Does it give you a new idea on how to approach the problem? If so, try it!

Hint

The word "denumerable" suggests natural numbers. Denumerable is another word for countably infinite.

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Solution

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A set A is denumerable if it is in bijective correspondence with the natural numbers. That is, it is denumerable if and only if there exists a bijective function
 $\displaystyle f:A\to \mathbb {N}$
Note: Mathematicians think of this function as a counting of all the elements of the set A. Given a counting, that is such a bijection ƒ the first element of the set A is the one that is mapped to 1 by the function, the second element is the one mapped to 2 and so on. Observe that we need to require the function to be bijective to guarantee that all the elements are counted and are counted in a unique way (we don't want to have two elements mapped to the same natural number).

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