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Let \( S \) be the universal set, where: \( S=\{1,2,3, \ldots, 18,19,20\} \) Let sets \( A \) and \( B \) be subsets of \( S \), where: Set \( A=\{1,3,4,7,9,11,12\} \) Set \( B=\{2,3,9,10,12,14,15,19,20\} \) Find the following: The number of elements in the set \( (A \cup B) \) : \( n(A \cup B)=1 \) The number of elements in the set \( (A \cap B) \) : \( n(A \cap B) \) is You may want to draw a Venn Diagram to help answer this question. Question Help: \( \square \) Video 1 Video 2

Ask by Knight Carrillo. in the United States
Mar 11,2025

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There are 13 elements in set \( A \cup B \) and 3 elements in set \( A \cap B \).

Solución

We start with the sets \[ A = \{1,3,4,7,9,11,12\} \quad \text{and} \quad B = \{2,3,9,10,12,14,15,19,20\}. \] **Step 1. Find \(A \cap B\):** \(A \cap B\) consists of the elements common to both \(A\) and \(B\). The common elements are: \[ 3,\; 9,\; 12. \] Thus, \[ n(A \cap B) = 3. \] **Step 2. Find \(A \cup B\):** \(A \cup B\) includes all elements that are in \(A\), in \(B\), or in both. Listing them in order, we have: \[ \{1,2,3,4,7,9,10,11,12,14,15,19,20\}. \] Counting these elements gives: \[ n(A \cup B) = 13. \] The final answers are: \[ n(A \cup B) = 13 \quad \text{and} \quad n(A \cap B) = 3. \]

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To find the number of elements in the sets \( A \cup B \) (the union of A and B) and \( A \cap B \) (the intersection of A and B), we can use the sets given. First, let's identify the elements of each set: - Set \( A = \{1, 3, 4, 7, 9, 11, 12\} \) - Set \( B = \{2, 3, 9, 10, 12, 14, 15, 19, 20\} \) **Union \( A \cup B \)** includes all unique elements from both sets: - Unique elements: \( \{1, 2, 3, 4, 7, 9, 10, 11, 12, 14, 15, 19, 20\} \) Counting these elements gives us: - \( n(A \cup B) = 13 \) **Intersection \( A \cap B \)** includes elements that are common to both sets: - Common elements are: \( \{3, 9, 12\} \) Counting these gives us: - \( n(A \cap B) = 3 \) So, the answers are: - \( n(A \cup B) = 13 \) - \( n(A \cap B) = 3 \)

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