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

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

UpStudy AI · Accepted Answer

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:
$$
n(A \cup B) = 17
$$
$$ n(A \cup B) = 17 $$

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

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