# Tagged Questions

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### Help to understand the proof of the cardinality of a Cartesian Product

I was reading the proof of this theorem: The proof is as follows: I understand the entire proof except for the conclusion: Because $h$ is bijective $\dots$ Why is it sufficient to show ...
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### How to prove two sets are mutually exclusive without using a Venn diagram?

If $E_1$ and $E_2$ are subsets of $S$, then $E_1$ and $E_2 \cap \overline{E_1}$ are mutually exclusive, where $\overline{E_1}$ is the complement of $E_1$. Is there a way to prove this without drawing ...
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### Cardinality of tautologies for propositional logic

I'm wondering how many tautologies there are in propositional logic. I'm thinking that it must be at least countable, since ($P_{1} \wedge P_{2} \wedge \cdots P_{n}) \models P_{i}$ should be a ...
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### Cardinality of Distinct Hilbert Systems with Detachment

Let us consider all formulas T of classical propositional logic which are tautologies up to simple substitution of variables where a variable can get simply substituted for another variable if and ...