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.

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.

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

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

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

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

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