# Tagged Questions

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### If each uncountable set $T$ has a countable subset, can we form $T$ by a union of countable subsets?

I was working my way through the set theory chapter in my Analysis textbook when I stumbled across these two theorems: Every infinite set has a countable subset A union of countable subsets ...
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### Enumeration of rational numbers

If $\Bbb Q=\{q_n:n\in \Bbb N\}$ be an enumeration of $\Bbb Q$, is it true that $|q_n|<1/n$ for infinitely many $n$? I just come up with this question, it seemed simple but I can't solve it. Is ...
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### Finite vs infinite distinction in Rudin's Analysis

I'm starting to self-study Rudin's Principles of Mathematical Analysis. I'm up to the second chapter, theorem 2.24. For any collection $\{G_i\}$ of open sets, $\bigcup_i^nG_i$ is open. For ...
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### Sets A and C with m-1 Elements

If A is a set with m elements and C is a set with one element, then A-C is a set with m-1 elements. What is a proof for this statement?
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### Proof Regarding Infinite Sets

If A is an infinite set and B is a finite set prove that A-B is an infinite set.
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### Would this function be correct?

I want to express a function that for any x input, it outputs the nearest EVEN integer less than or equal to x. Would $g(x) = \{ \lfloor x \rfloor : 2 \mid x \}$ do the job properly? Read: $g(x)$ ...
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### Suppose $f: X \to Y$ and $g: Y \to Z$ are onto functions. Show that $g \circ f: X \to Z$ is onto

I feel like my solution isn't sufficient at the moment. --> For some x in X there exists a y in Y given by $f(x)=y$ because $f: X \to Y$ is onto. The same can be said for some y in Y existing as ...
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