Inclusion/Exclusion

absvalue · #1 $Old$ November 3rd, 2009, 10:19 AM

I've been working on this problem for a while. Could someone show me how to connect the last part?

Let A, B, and C be finite sets. Prove:
If $|A \cup B \cup C| = |A| + |B| + |C|$
then A, B, and C must be pairwise disjoint.

Here is what I have:

Suppose A, B, and C are finite sets with $|A \cup B \cup C| = |A| + |B| + |C|$ .

By inclusion/exclusion, we know that

$|A \cup B \cup C| = |A| + |B| + |C| - |A \cap B| - |A \cap C| - |B \cap C| + |A \cap B \cap C|$

By cancellation, we have:

$|A \cap B \cap C| - |A \cap B| - |A \cap C| - |B \cap C| = 0$

I'm just not sure how to connect that to "Thus A, B, and C must be pairwise disjoint."

gmatt · #2 $Old$ November 3rd, 2009, 10:46 AM

Quote:

Originally Posted by absvalue $View Post$

I've been working on this problem for a while. Could someone show me how to connect the last part?

Let A, B, and C be finite sets. Prove:
If $|A \cup B \cup C| = |A| + |B| + |C|$
then A, B, and C must be pairwise disjoint.

Here is what I have:

Suppose A, B, and C are finite sets with $|A \cup B \cup C| = |A| + |B| + |C|$ .

By inclusion/exclusion, we know that

$|A \cup B \cup C| = |A| + |B| + |C| - |A \cap B| - |A \cap C| - |B \cap C| + |A \cap B \cap C|$

By cancellation, we have:

$|A \cap B \cap C| - |A \cap B| - |A \cap C| - |B \cap C| = 0$

I'm just not sure how to connect that to "Thus A, B, and C must be pairwise disjoint."