True or False: Countable vs. Uncountable Sets

Question

Determine if the statement is true or false. If true provide a proof or give a counterexample if the statement is false.

Any set $S$ containing a countable set $T$ must be uncountable.

Can someone help me to see whether the statement is true or false?

Part of me wants to say the statement is true. If we let the set S be the set of real numbers and T be the set of rational numbers. Does this proof satisfy the generalized statement though?

Answer 1

There are a few things to draw out from this.

1. Your answer is insufficient. You've been asked to prove "if A is always true, then statement S is always true"; you've exhibited an example of an A which satisfies statement S; and so you've deduced that S is always true. This just isn't valid as a step of reasoning: "all men are Vladimir Putin" is not true just because I can show you a man who is Vladimir Putin.
2. You should be crystal clear about what it means for a set to "contain" another. A set $S$ contains an element $x$ if $x \in S$. You're intending the following usage instead: a set $S$ contains a subset $T$ if $T \subseteq S$. This is clear from context, and what you've said is not ambiguous except if someone is reading with an eye to spotting ambiguity; but it looks like you're in the stage of your mathematical career when it pays to be absolutely clear about this kind of thing. I'd prefer it if you said "Any set $S$ which has a countable subset $T$ must be uncountable". Note that the set $S = \{\mathbb{N}\}$ is finite but contains a countably infinite set in the sense that there's a countably infinite set $\mathbb{N}$ which is a member of $S$; but it doesn't contain an infinite subset.

To show that the "theorem" is true, you're going to have to prove that whenever your worst enemy hands you a set $S$ and a countably infinite subset $T$ of $S$, then $S$ is uncountably infinite (however maliciously your enemy chose $T$ and $S$).

To show that the "theorem" is false, it's enough to find a single set $S$ with a single countably infinite subset $T$, such that $S$ is not uncountable.

Hint: every set is a subset of itself.