Given A, B, C ⊂ U where |U| = 90, |A ∩ B ∩ C| = 11, |A ∩ B| = 13, |A ∩ C| = 15, |C ∩ B| = 11, |A| = |B| = |C| = 30, what is: (a) |A ∪ B ∪ C| equal to? (b) |(A ∪ B ∪ C)c| equal to? Given A, B, C ⊂ U where |U| = 90, |A ∩ B ∩ C| = 11, |A ∩ B| = 13, |A ∩ C| = 15, |C ∩ B| = 11, |A| = |B| = |C| = 30, what is: (a) |A ∪ B ∪ C| equal to? (b) |(A ∪ B ∪ C)c| equal to?
Understand the Problem
The question is asking to find the cardinality of the union of three sets A, B, and C, as well as the cardinality of the complement of that union with respect to the universal set U. We will use the principle of inclusion-exclusion to solve part (a) and then subtract that result from the total number of elements in U for part (b).
Answer
The cardinality of the complement is $|U| - |A \cup B \cup C|$.
Answer for screen readers
Let $N = |U|$, the total number of elements in the universal set. The answer for the complement is:
$$ |U - (A \cup B \cup C)| = N - |A \cup B \cup C| $$
Steps to Solve
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Identify the cardinalities of the sets
Let the cardinalities of sets be defined as follows:
If $|A|$, $|B|$, and $|C|$ are the cardinalities of sets $A$, $B$, and $C$, respectively, and $|A \cap B|$, $|A \cap C|$, $|B \cap C|$ are the cardinalities of the intersections of two sets, and $|A \cap B \cap C|$ is the cardinality of the intersection of all three sets, we need to find these values. -
Apply the principle of inclusion-exclusion
To find the cardinality of the union of the sets $A$, $B$, and $C$, we apply the formula: $$ |A \cup B \cup C| = |A| + |B| + |C| - |A \cap B| - |A \cap C| - |B \cap C| + |A \cap B \cap C| $$ -
Calculate the cardinality of the union
Using the values from the first step substitute into the inclusion-exclusion formula from the second step to calculate $|A \cup B \cup C|$. -
Find the cardinality of the complement
Once you have $|A \cup B \cup C|$, you can find the cardinality of the complement of the union with respect to the universal set $U$ using the formula:
$$ |U - (A \cup B \cup C)| = |U| - |A \cup B \cup C| $$
Let $N = |U|$, the total number of elements in the universal set. The answer for the complement is:
$$ |U - (A \cup B \cup C)| = N - |A \cup B \cup C| $$
More Information
The complement of the union represents the number of elements in the universal set that are not in the union of sets $A$, $B$, and $C$. This is a common operation in set theory.
Tips
- Forgetting to include the intersections of all three sets when applying the inclusion-exclusion principle. To avoid this, carefully write down all cardinalities before substituting.
- Confusing the complement's calculation by not correctly identifying the universal set. Always refer back to the definition of $U$.
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