Prove that a disjoint union of any finite set and any countably infinite set is countabley infinite.
A disjoint union of any finite set and any countably infinite set is countably infinite.
A disjoint union of any finite set and any countably infinite set.
Let be a finite set and let B be a countably infinite set such that A and B are disjoint . We then need to proof that is countably infinite.
Since B is countable infinite, B has the same cardinality as Z + and thus there exists a one-to-one correspondence f between B and Z + (which implies that f is onto and one-to-one).
Let us define the function g as:
If y > n, then there exists some elements such that (as f is one-to-one and
y − n > 0).
We then note that for every there exists an element or (respectively) such that
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