\( \left\{\begin{array}{l}n(a)=8 \\n(b)=12 \\n(anb)=3 \\find \\n(aub)using \\venndiagram\end{array}\right. \)

To find \( n(A \cup B) \) using a Venn diagram, we can use the principle of inclusion-exclusion. Here's a step-by-step explanation: ### Given: - \( n(A) = 8 \)   (Number of elements in set \( A \)) - \( n(B) = 12 \)   (Number of elements in set \( B \)) - \( n(A \cap B) = 3 \)   (Number of elements common to both sets \( A \) and \( B \)) ### Objective: Find \( n(A \cup B) \)   (Number of elements in either set \( A \), set \( B \), or both) ### Formula: The number of elements in the union of two sets is given by: \[ n(A \cup B) = n(A) + n(B) - n(A \cap B) \] ### Calculation: Plugging in the given values: \[ n(A \cup B) = 8 + 12 - 3 = 17 \] ### Visualization with a Venn Diagram: Imagine two overlapping circles representing sets \( A \) and \( B \): - **Set \( A \)** has 8 elements. - **Set \( B \)** has 12 elements. - The **overlapping region** (common to both sets) has 3 elements. To find the total number of unique elements in either set: 1. **Add all elements from both sets**: \( 8 + 12 = 20 \) 2. **Subtract the overlapping elements** (since they've been counted twice): \( 20 - 3 = 17 \) ### Final Answer: \[ n(A \cup B) = 17 \]