Work in a reasonable theory of determinacy such as $\mathsf{ZF+DC+AD}$. Which of the following identities are true for arbitrary infinite sets?

- $|A^2|=|A^3|$ (motivated by an MSE question that asks if this identity implies choice; a well known theorem of Tarski says $|A|=|A^2|$ implies choice)
- $|A|=|A\sqcup A|$
- $|A\times\omega|=|A|$

More generally, is there a procedure to determine the validity of any cardinality identity involving $A$, $\omega$, $\times$ and $\sqcup$?

**Second try:** As pointed out by Asaf Karagila, the answer is negative for a somewhat stupid reason...Here are some attempts to make the question more interesting:

- Work in $\mathsf{ZF+AD}+V=L(\mathbb{R})$, or determine the truth in $L(\mathbb{R})$ under large cardinals (are these the same?)
- Only consider "small" infinite sets $A$ (such as images of $\mathbb{R}$).

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