- Identify the given sets: \(U = \{1, 2, 3, 4, 5, 6, 7, 8, 9\}\), \(A = \{1, 6, 7, 8, 9\}\), \(B = \{1, 3, 4, 7, 9\}\), \(C = \{4, 5, 6, 7\}\).
- Use De Morgan's Law to simplify the expression: \(C' \cup (A \cap B)' = (C \cap A \cap B)'\).
- Calculate the intersection \(C \cap A \cap B = \{7\}\).
- Find the complement of this set relative to \(U\) to get the final result: \(\boxed{\{1, 2, 3, 4, 5, 6, 8, 9\}}\).
Explanation
- Analysis of the sets and objective
We are given the universal set \(U = \{1, 2, 3, 4, 5, 6, 7, 8, 9\}\) and three subsets:
\(A = \{1, 6, 7, 8, 9\}\)
\(B = \{1, 3, 4, 7, 9\}\)
\(C = \{4, 5, 6, 7\}\)
Our objective is to find the set \(C' \cup (A \cap B)'\). We recall that for any two sets \(X\) and \(Y\), the complement of their intersection is the union of their complements, known as De Morgan's Law: \((A \cap B)' = A' \cup B'\). Thus, the expression simplifies to \(C' \cup A' \cup B'\), which by further application of De Morgan's Law is equivalent to \((C \cap A \cap B)'\).
- Calculating the set expression using De Morgan's Law
First, we compute the intersection \(A \cap B\):
\(A \cap B = \{1, 6, 7, 8, 9\} \cap \{1, 3, 4, 7, 9\} = \{1, 7, 9\}\)
Next, we find the intersection \(C \cap (A \cap B)\):
\(C \cap (A \cap B) = \{4, 5, 6, 7\} \cap \{1, 7, 9\} = \{7\}\)
Now, we find the complement of this intersection within the universal set \(U\):
\((C \cap A \cap B)' = U \setminus \{7\} = \{1, 2, 3, 4, 5, 6, 8, 9\}\)
- Verification and Conclusion
Alternatively, we calculate each component independently to verify the result:
The complement of \(C\) is \(C' = \{1, 2, 3, 4, 5, 6, 7, 8, 9\} \setminus \{4, 5, 6, 7\} = \{1, 2, 3, 8, 9\}\).
The complement of \(A \cap B\) is \((A \cap B)' = \{1, 2, 3, 4, 5, 6, 7, 8, 9\} \setminus \{1, 7, 9\} = \{2, 3, 4, 5, 6, 8\}\).
Taking the union of these two sets:
\(C' \cup (A \cap B)' = \{1, 2, 3, 8, 9\} \cup \{2, 3, 4, 5, 6, 8\} = \{1, 2, 3, 4, 5, 6, 8, 9\}\).
Both methods yield the same result.
Examples
Understanding set operations is essential in database management and search systems. For instance, when a librarian searches for books that are either "not in category C" OR "not in the intersection of categories A and B", they are essentially applying the set logic demonstrated here. By calculating the complement of these combined categories, the system can quickly exclude irrelevant items and present the user with the correct subset of the library's collection.