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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

enter image description here

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The 'proof' you show in your link using Venn Diagrams is pretty good and well organized, but maybe you need to have some more explanations to go with the pictures.

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.) $$(A \cap C) \cap (B \cap C) = A \cap B \cap C \cap C\\ = A \cap B \cap C = (A \cap B) \cap C\\ = \emptyset \cap C = \emptyset.$$

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