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If $A$ and $B$ are infinite sets, is it true that the cardinality of its Cartesian product ($|A \times B|$) is equal to max $(|A|,|B|)$? If it is true, why? (We assume the axiom of choice)

Andrés E. Caicedo
  • 79,201

1 Answers1

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Correct me please if I'm not right.

  1. $|A \times B| \geq \max (|A|,|B|)$

  2. $ |A \cup B| = \max (|A|, |B|)$

  3. $| (A \cup B) \times (A \cup B)| = |A \cup B| = \max (|A|, |B|)$

  4. $|A \times B| \leq | (A \cup B) \times (A \cup B)| = \max (|A|, |B|)$

  5. $\max (|A|, |B|) \leq |A \times B| \leq \max (|A|, |B|)$

then $|A \times B| = \max (|A|, |B|)$

Chickenmancer
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  • The claim that $|(A\cup B)\times(A\cup B)|=|A\cup B|$ (and more generally, that for an infinite set $C$, $C\times C$ is bijectable with $C$, is equivalent to the Axiom of Choice (theorem of Tarski) – Arturo Magidin May 06 '20 at 21:47