Find the number of elements in $A \cup B \cup C$ if there are 50 elements in $A$, 500 in $B$, and 5,000 in $C$

Question

I am given this:

Find the number of elements in $A \cup B \cup C$ if there are 50 elements in $A$, 500 in $B$, and 5,000 in $C$ if:

1. $A \subseteq B$ and $B \subseteq C$

2. The sets are pairwise disjoint.

3. There are two elements common to each pair of sets and one element in all three sets

For #1, would the answer simply be $A \cap B$ and $B \cap C$, given that if A is a subset of B, and B is a subset of C, then A and B having common elements (as well as B and C), then it would be the union of the two sets?

As for the other two, I am a little confused by the wording and could use some help on those.

Answer 1

"Pairwise disjoint" means that the number of elements in $A \cup B \cup C$ equals the sum of the number of elements in $A,$ $B,$ and $C$ counted distincly. This is simply $5000 + 500 + 50 = \boxed{5550}.$

Use PIE on the last part. We find the answer as follows: $$(5000 + 500 + 50) - (2 + 2 + 2) + 1$$ $$= \boxed{5545}.$$

Answer 2

1. If $A\subseteq B \subseteq C$, then $A\cup B\cup C = C$.
2. In general, $|A\cup B\cup C|=|A|+|B|+|C|-|A\cap B|-|B\cap C| -|C\cap A|+|A\cap B\cap C|$. If sets are pairwise disjoint, what will happen? And how about third question?

Answer 3

I remember this question... here is an excerpt from my old textbook. Note that the threes in the last line would simply be twos because your question differed in what it was asking by just that.