Let's match the formulas with the corresponding Venn diagrams by analyzing both the set operations and the shaded areas.

### Formula 1: \( M \cup N = N \cup M \)
- **Explanation**: This is the **commutative property** of the union. The union of \(M\) and \(N\) includes all elements that belong to either set. 
- **Matching Diagram**: Look for a Venn diagram where both the regions of \(M\) and \(N\) are fully shaded. The position of the sets doesn't matter because of the commutative property.

### Formula 2: \( M \cup K = K \cup M \)
- **Explanation**: Again, the **commutative property**, but this time between sets \(M\) and \(K\). The union should include all areas covered by either set \(M\) or set \(K\).
- **Matching Diagram**: Look for a Venn diagram where both the regions of \(M\) and \(K\) are fully shaded.

### Formula 3: \( K \cup N = N \cup K \)
- **Explanation**: Another **commutative property** between \(K\) and \(N\). The union includes all elements that belong to either set \(K\) or \(N\).
- **Matching Diagram**: Look for a Venn diagram where the regions of \(K\) and \(N\) are fully shaded.

### Formula 4: \((M \cup K) \cup N = M \cup (K \cup N)\)
- **Explanation**: This is the **associative property** of union. The union of all three sets should be the same, no matter how you group them.
- **Matching Diagram**: The Venn diagram for this formula should show the union of all three sets \(M\), \(K\), and \(N\), meaning the entire area covered by all three circles should be shaded.

### Now, matching each formula with the diagrams:

1. **\( M \cup N = N \cup M \)**: This matches with the **first diagram** (top-right), where both \(M\) and \(N\) are shaded.
2. **\( M \cup K = K \cup M \)**: This corresponds to the **second diagram** (second row, right), where both \(M\) and \(K\) are shaded.
3. **\( K \cup N = N \cup K \)**: This matches the **third diagram** (third row, right), where \(K\) and \(N\) are shaded.
4. **\((M \cup K) \cup N = M \cup (K

Explore how to match set theory formulas with corresponding Venn diagrams. This solution demonstrates key concepts like the commutative and associative properties of set operations, along with detailed step-by-step explanations. Ideal for students in Grades 9-12.

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