- #1
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- TL;DR Summary
- It seems obvious that ##A \subset B## implies ##|A|<|B|##. Does it need proof?
Hi,
I gave my friend a proof that the set of pairs ##\ mathbb N \times \mathbb N \times...\mathbb N ## (Finite product) is countable. I gave an injection :
##(a_1, a_2,...,a_n) \right arrow p_1_^{a_1} p_2^{a_2}...p_n^{a_n}##
Where the ##p_i## are distinct primes. My friend is telling me this is not enough and I can't see why,but I may be missing some foundational issues. I should each such product injects into the Naturals and any subset of the Naturals is countable. Am I missing anything here? Seems he's missing something or blowing it out of proportion.
I gave my friend a proof that the set of pairs ##\ mathbb N \times \mathbb N \times...\mathbb N ## (Finite product) is countable. I gave an injection :
##(a_1, a_2,...,a_n) \right arrow p_1_^{a_1} p_2^{a_2}...p_n^{a_n}##
Where the ##p_i## are distinct primes. My friend is telling me this is not enough and I can't see why,but I may be missing some foundational issues. I should each such product injects into the Naturals and any subset of the Naturals is countable. Am I missing anything here? Seems he's missing something or blowing it out of proportion.