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In textbook Algebra, Hungerford I read this proof:

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I don't understand the part in yellow which refers to this theorem:

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asv
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  • So for cardinal numbers (also infinite) if a+b=c+b then a=c? – asv Dec 29 '19 at 19:23
  • Counterexample to cancellation law in cardinals addition:

    https://math.stackexchange.com/questions/144051/counterexample-to-cancellation-law-in-cardinals-addition

     – asv Dec 29 '19 at 20:45
  • I'm not using a cancellation law, I am directly applying Theorem 8.10. Though I should have said that $|S-F|+|F| = |S-F|$ by Theorem 8.10, and so since we also have $|S-F|+|F|= |S|$, the result follows. – xxxxxxxxx Dec 29 '19 at 22:32
  • So you put $|S-F|= \alpha$ and $|F|= \beta$ but why $\alpha$ is infinite and $\beta \leq \alpha$? – asv Dec 30 '19 at 09:57
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    You choose in the beginning an $S$ with $\aleph_{0}|K| < |S|$ so $S$ is infinite. Then since $|F| < |S|$, $|S-F|$ must still be infinite. Also if $|F| > |S-F|$ then this would imply that $|S-F| + |F| = |F|$ (using Theorem 8.10), which would imply that $|F| \geq |S|$. – xxxxxxxxx Dec 30 '19 at 17:07
  • Ok, thank you very much. – asv Dec 30 '19 at 18:28

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