Murray said, “But the odd integers can be derived. If you divide every even integer in the entire infinite series by two, you will get another infinite series which will contain within it the infinite series of odd integers.”
For S={2,4,6,8...}
S/2={1,2,3,4,..}
I used to think the set of even integers is a subset of the natural numbers.Doesnt this suggest that the reverse (the set of natural numbers being a subset of even integers) is actually true?
I think what you mean is you used to think that the set of non-negative even integers was "smaller" than the set of natural numbers. The reason you can do this is because the set of non-negative even integers has the same cardinality as the set of naturals (both are countably infinite): http://en.wikipedia.org/wiki/Cardinality
The same is true of the rational numbers, by the way. There is a famous proof of the fact that there are infinite sets with larger cardinality that the naturals (the reals for example): http://en.wikipedia.org/wiki/Cantor's_diagonal_argument
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Murray said, “But the odd integers can be derived. If you divide every even integer in the entire infinite series by two, you will get another infinite series which will contain within it the infinite series of odd integers.”
For S={2,4,6,8...}
S/2={1,2,3,4,..}
I used to think the set of even integers is a subset of the natural numbers.Doesnt this suggest that the reverse (the set of natural numbers being a subset of even integers) is actually true?
I think what you mean is you used to think that the set of non-negative even integers was "smaller" than the set of natural numbers. The reason you can do this is because the set of non-negative even integers has the same cardinality as the set of naturals (both are countably infinite): http://en.wikipedia.org/wiki/Cardinality
The same is true of the rational numbers, by the way. There is a famous proof of the fact that there are infinite sets with larger cardinality that the naturals (the reals for example): http://en.wikipedia.org/wiki/Cantor's_diagonal_argument
No, not a subset, for no more reason than that 17 is a member of the latter set and not of the former one.
What you're seeing is a 1:1 correspondence between an infinite set (the natural numbers) and a proper subset of same (the even natural numbers).