# Show that if A and B are disjoint, then A ∩ C and B ∩ C are also disjoint

I need to show using Venn diagram. Is my solution correct, I understand from the Venn diagram it would be right, but can someone explain in a more formal/mathematical way why this implication is true?

https://i.stack.imgur.com/ChIFX.jpg

Sets $$A \cap C$$ and $$B \cap C$$ are disjoint, if their intersection is null. According to rules of sets in your text or lecture notes, can you supply a justification for each $$=$$-sign below? (Add or delete steps as appropriate.) $$(A \cap C) \cap (B \cap C) = A \cap B \cap C \cap C\\ = A \cap B \cap C = (A \cap B) \cap C\\ = \emptyset \cap C = \emptyset.$$