Does $|A \cap B|+|C| = |A \cap C|+|B \cap C|$ conclude $A \cap B \subseteq C \subseteq A\cup B$?

Question

Suppose $A,B,C$ are finite subsets of a universal set $\mathcal{M}$

Could someone infer from $|A \cap B|+|C| = |A \cap C|+|B \cap C|$ that

$A \cap B \subseteq C \subseteq A\cup B$ ?

Answer 1

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 $$A\cap B\subseteq C \subseteq A\cup B$$