Questions: Determine if the following statement is true or false. It can be proved that (A ∩ B)'=A' ∪ B', so this means that the complement of the intersection of two sets is the union of the complement of those sets. Choose the correct answer below. A. The statement is false because (A ∩ B)'=A' ∪ B' means that the complement of the intersection of two sets is the intersection of the complement of those sets. B. The statement is false because a theorem is only sometimes true. C. The statement is true because a theorem is always true and (A ∩ B)'=A' ∪ B' means that the complement of the intersection of two sets is the union of the complement of those sets. D. The statement is false because (A ∩ B)'=A' ∪ B' means that the complement of the union of two sets is the union of the complement of those sets.

Determine if the following statement is true or false. It can be proved that (A ∩ B)'=A' ∪ B', so this means that the complement of the intersection of two sets is the union of the complement of those sets. Choose the correct answer below. A. The statement is false because (A ∩ B)'=A' ∪ B' means that the complement of the intersection of two sets is the intersection of the complement of those sets. B. The statement is false because a theorem is only sometimes true. C. The statement is true because a theorem is always true and (A ∩ B)'=A' ∪ B' means that the complement of the intersection of two sets is the union of the complement of those sets. D. The statement is false because (A ∩ B)'=A' ∪ B' means that the complement of the union of two sets is the union of the complement of those sets.
Transcript text: Determine if the following statement is true or false. It can be proved that $(A \cap B)^{\prime}=A^{\prime} \cup B^{\prime}$, so this means that the complement of the intersection of two sets is the union of the complement of those sets. Choose the correct answer below. A. The statement is false because $(A \cap B)^{\prime}=A \prime \cup B^{\prime}$ means that the complement of the intersection of two sets is the intersection of the complement of those sets. B. The statement is false because a theorem is only sometimes true. C. The statement is true because a theorem is always true and $(A \cap B)^{\prime}=A^{\prime} \cup B^{\prime}$ means that the complement of the intersection of two sets is the union of the complement of those sets. D. The statement is false because $(A \cap B)^{\prime}=A^{\prime} \cup B^{\prime}$ means that the complement of the union of two sets is the union of the complement of those sets.
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Solution

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Solution Steps

To determine if the statement is true or false, we need to verify if the given expression \((A \cap B)^{\prime} = A^{\prime} \cup B^{\prime}\) is correct. This is a known result from set theory called De Morgan's Law, which states that the complement of the intersection of two sets is the union of their complements. Therefore, the statement is true.

Step 1: Define Sets

Let \( A = \{1, 2, 3\} \) and \( B = \{3, 4, 5\} \).

Step 2: Calculate the Complement of the Intersection

The intersection \( A \cap B = \{3\} \). The complement of the intersection, denoted as \( (A \cap B)^{\prime} \), is calculated as: \[ (A \cap B)^{\prime} = \{1, 2, 4, 5\} \]

Step 3: Calculate the Complements of Each Set

The complement of set \( A \), denoted as \( A^{\prime} \), is: \[ A^{\prime} = \{4, 5\} \] The complement of set \( B \), denoted as \( B^{\prime} \), is: \[ B^{\prime} = \{1, 2\} \]

Step 4: Calculate the Union of the Complements

The union of the complements \( A^{\prime} \cup B^{\prime} \) is: \[ A^{\prime} \cup B^{\prime} = \{1, 2\} \cup \{4, 5\} = \{1, 2, 4, 5\} \]

Step 5: Compare the Results

We find that: \[ (A \cap B)^{\prime} = \{1, 2, 4, 5\} \quad \text{and} \quad A^{\prime} \cup B^{\prime} = \{1, 2, 4, 5\} \] Since both expressions are equal, we conclude that the statement \( (A \cap B)^{\prime} = A^{\prime} \cup B^{\prime} \) is true.

Final Answer

The answer is \( \boxed{C} \).

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