Cole Zimmerman
06/22/2024 · Elementary School
* If \( U=\{u, v, w, x, y, z\}, A=\{u, v, y\} \) and \( B=\{x, y, z\} \), then prove that \( (A \cup B B)^{C}=A^{C} \cup B \).
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Tutor-Verified Answer
Step-by-step Solution
To prove that \( (A \cup B B)^{C}=A^{C} \cup B \), we need to show that the complement of the union of sets \( A \) and \( B \) is equal to the union of the complement of set \( A \) and set \( B \).
Given:
- \( U=\{u, v, w, x, y, z\} \)
- \( A=\{u, v, y\} \)
- \( B=\{x, y, z\} \)
Let's start by finding the union of sets \( A \) and \( B \):
\[ A \cup B = \{u, v, y\} \cup \{x, y, z\} = \{u, v, x, y, z\} \]
Now, let's find the complement of the union of sets \( A \) and \( B \):
\[ (A \cup B)^{C} = \{u, v, x, y, z\}^{C} = \{w\} \]
Next, let's find the complement of set \( A \):
\[ A^{C} = \{u, v, y\}^{C} = \{w, x, z\} \]
Finally, let's find the union of the complement of set \( A \) and set \( B \):
\[ A^{C} \cup B = \{w, x, z\} \cup \{x, y, z\} = \{w, x, y, z\} \]
Therefore, we have shown that \( (A \cup B B)^{C}=A^{C} \cup B \).
Quick Answer
To prove \( (A \cup B)^{C}=A^{C} \cup B \), find the union of \( A \) and \( B \), then find the complements of both sides. The result shows they are equal.
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