# Let 𝐴, 𝐵 and 𝐶 be sets. Prove formally that |𝐴 ∪ 𝐵 ∪ 𝐶| = |𝐴| + |𝐵| + |𝐶| − |𝐴 ∩ 𝐵| − |𝐴 ∩ 𝐶| − |𝐵 ∩ 𝐶| + |𝐴 ∩ 𝐵 ∩ 𝐶|

By using a Venn diagram we can see almost immediately that the cardinality of the members of the equality is in fact the same, however the exercise asks me to prove it formally and there is where my problem lies.

I have though on making a bijection to prove it, I just don't see how I can do it when I am not given any more information in the statement, can someone guide me to what I have to do?

Thanks a lot.

• To formalize the argument "each part of the Venn diagram except the intersection is cancelled, and the intersection cancels except for one term": You could write $|A| = |A \cap B \cap C| + |A \cap B^c \cap C| + |A \cap B \cap C^c| + |A \cap B^c \cap C^c|$ and similarly $|A \cap B| = |A\cap B\cap C| + |A\cap B\cap C^c|$. – Daniel Schepler Nov 21 '18 at 22:44

HINT: First prove that $$|A\cup B|=|A|+|B|-|A\cap B|$$, and then use this and the fact that $$\cup$$ and $$\cap$$ are associative to prove the formula for $$|A\cup B\cup C|$$.