Suppose $A,B,C$ Are finite subsets of a universal set $\mathcal{ M}$
Can someone infer from $|A \cap B|+|C| = |A \cap C|+|B \cap C|$ that it must be
$A \cap B \subseteq C \subseteq A\cup B$ ?
What if $|A \cap B|+|C| = |A \cap C|+|B \cap C|$
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elementary-set-theory
1 Answers
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Hint: Draw a Venn diagram, name the 7 areas accordingly, understand what the equation means exactly. The equation says, $$C\setminus (A\cup B)= \emptyset$$ $$(A\cap B) \setminus C= \emptyset$$ For example :
$$|A\cap B|= AB+ABC$$ $$|C| = C+BC+AC+ABC$$ $$|A\cap C|= AC+ABC$$ $$|B\cap C|= BC+ABC$$ So, the equation means, $AC+BC+ABC+ABC=C+BC+AC+ABC+AB+ABC$, or, $AB+C=0$. So, indeed $$A\cap B\subseteq C \subseteq A\cup B$$