I have a problem to show that the symmetric difference of sets A and C is a subset of the union of the symmetric differences of sets A and B and sets B and C.

\( A \Delta C \subseteq (A \Delta B) \cup (B \Delta C) \)

From the definition of the symmetric difference, really rewritten

\( (A \cap C^c) \cup (A^c \cap C) \subseteq (A \cap B^c) \cup (A^c \cap B) \cup (B \cap C^c) \cup (B^c \cap C) \)

Using distributive laws

\( (A \cap C^c) \cup (A^c \cap C) \subseteq (B^c \cap (A \cup C)) \cup (B \cap (A^c \cup C^c)) \)

At this point I am unsure how to proceed. My proof strategy is attempting to show for some element in the subset necessarily implies that is an element of the superset, by rewriting the superset in a form which will make this apparent.

Thanks for reading,