I need to prove that the set S of all finite subsets of ℕ with exactly n elements (where n is a natural number) is denumerable. I think I need to set up an injection from S to ℕ since ℕ is a countable set but am not sure how exactly to describe that 1-1 mapping? I understand the proof of the set of all finite sets being denumerable but I'm just struggling when you specify that the cardinality of the finite sets has to be n.
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One approach among many: For two elements, ${0,1},{0,2},{1,2},{0,3},{1,3},{2,3},{0,4},{1,4},{2,4},{3,4},{0,5},\dots$. Can you formalize this? Can you generalize this to $n>2$, more than two elements at a time? – JMoravitz Apr 30 '21 at 13:53
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A subset of a denumerable set is denumerable, right, mathhelp? – Gerry Myerson Apr 30 '21 at 13:55