I've encountered these two statements below, and I fail to understand why one of them is true and the other isn't. Let $X_1,X_2$ be groups of propositions,and $Z_1,Z_2$ be their satisfying groups of valuation then: $Z_1 \cap Z_2$ satisfies $X_1 \cap X_2$ (is false according to the text book),the other statement is $Z_1 \setminus Z_2$ satisfies $X_1 \setminus X_2$ (which is true according to the text book). Every counter argument I tried to come up with also counters the "true" statement so I'm lost and would gladly use some help.
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Here's a MathJax tutorial :) – Shaun Feb 09 '20 at 16:53
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Thanks,I'll read that later. – Netanel Ron Feb 09 '20 at 17:03
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What textbook ? – Mauro ALLEGRANZA Feb 09 '20 at 17:11
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Local one written by our professor, it might be false therefore I'm trying to get answers from the internet too just to make sure. – Netanel Ron Feb 09 '20 at 17:20
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Without further details, IMO you are right: the first is clearly false. A tautology is satisfied by every valuation; thus, every tautology must belong to both $X_1$ and $X_2$. – Mauro ALLEGRANZA Feb 09 '20 at 17:21
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That's what I had in mind, thank you for validating me. – Netanel Ron Feb 09 '20 at 17:24