1. Let S and T be sets. Show that |S| ≤ |T| if and only if there...

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1. Let S and T be sets. Show that |S| ≤ |T| if and only if there is a surjective map from T onto S.

2. Let (Sn) be a sequence of countable sets. Show that U Sn is countable.

3. Show that N₀ ≤ k for each infinite cardinal k.

4. A Hamel basis of a (possibly infinite dimensional) vector space (over an arbitrary field) is a linearly independent subset whose linear span is the whole space. Use Zorn's lemma to show that every non zero vector space has a Hamel basis.

5. Prove that |W| x |R| = C.
(Hint: First use the Contor-Bernstein Theorem to prove that |[0, 1)| = |[0, ∞)| = |R|.
Then show that |W| x |R| = |W x [0, 1)| = |[0, ∞).)

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