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Math Help - question on cardinality and showing that the set of integers is countable

  1. #1
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    question on cardinality and showing that the set of integers is countable

    i know that two sets have the same cardinality if there exists a bijective correspondence between them and that a set is countable if there exists a bijective correspondence between that set and the set of natural numbers. My teacher proved the set of integers was countable by listing the positive integers along with 0. Then he said that he could just sandwich the negative integers in between the positive ones so the list would look like 0 -1 1 -2 2 -3 3... and so on. What i'm not very clear on is why my teacher chose to sandwich the negative integers between the positive integers like that. couldn't he just have tacked the negative integers at the end of the list of positive integers?
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  2. #2
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    The "end of the list of positive integers"? Where is the "end"? The set of positive integers is infinite. A list of them does NOT have an "end".
    Every integer is either 0, positive or negative. You can map them into the positive integers by:
    f(0)= 1

    f(n)= 2n+ 1 if n is positive

    f(n)= -2n if n is negative.

    That is, 0 is mapped to 1, the positive integers are mapped to the odd positive integers in order and the negative integers are mapped to the even positive integers.
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