[Math] Suppose $A, B$, and C are sets. Prove that $C\subset A\Delta B \Leftrightarrow C \subset A \cup B$ and $A \cap B \cap C = \emptyset $

elementary-set-theorylogic

The problem statement is in the title.

I'm proving a problem in class and I need to show the above containment. I've drawn some Venn diagrams to make sure the containment makes sense, and it does to me, but I am still having problems proving this statement.
I know that $A \Delta B = (A \cup B) – ( A \cap B)$, and I can see through the Venn diagrams I have drawn how [$C \subset A \Delta B$ iff $C \subset A \cup B$ and $A \cap B \cap C =\emptyset $], but I am having trouble explaining this to someone else.

Best Answer

Let's try necessity. Let $$c \in C \subseteq A \Delta B = (A \cup B) - (A \cap B).$$ Then, $c \in A \cup B$ and $c \not \in A \cap B$. Thus,

  • every $c \in C$ also satisfies $c \in A \cup B$, hence, $C \subseteq A \cup B$;
  • every $c \in C$ also satisfies $c \not \in A \cap B$, hence $C \cap A \cap B = \emptyset$.

Can you go the other way?

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