Let A and B be countable infinite sets. Being both countable, a one-to-one correspondence between the set’s elements can be established. A new correspondence can also be established between A and the union of all elements in A and B, since this is another countable set.
Intuitively, it would seem even more obvious that the same would apply if A was uncountable. The larger set would still be uncountable. However, the countability is part of the proof in the first case, such as two car lanes merging into one. The same technique is not available in the second case.
(I note the question: An uncountable set minus a countable set is still uncountable, but is that equivalent to my question? – It would be if there is a one-to-one correspondence between all uncountable sets, but I don't think this is obvious)
How is such a proof delivered?
Best Answer
We can see that an uncountable set minus a countable set is indeed uncountable.
Suppose for an uncountable set $A$ and a countable set $B$ that $A-B$ is countable. The union of countably many countable sets is countable; thus $(A-B) \cup B$ is countable. But then A is a subset of $(A-B) \cup B$ and thus must be countable itself, which is a contradiction.
It is however not true that there is a bijection between all uncountable sets. The cardinality of the continuum is one such cardinality of an uncountable set, but by Cantor's theorem, the power set of the reals has a cardinality strictly larger than that of the continuum. The power set of the power set of the reals has yet a larger cardinality. Thus there are infinitely many uncountable cardinalities, by extension.
Does this help?
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EDIT: see this also.