https://file.aiquickdraw.com/mathbot/file/1729870013992fifhyxul.jpg

The image asks for the interpretation of set operations using Venn diagrams for three sets \(X\), \(Y\), and \(Z\). The two set expressions to evaluate are:

1. \((X \cap Y) \times Z\)
2. \((X \cup Y) \times (Z - Y)\)

Here’s a breakdown of each:

1. **\((X \cap Y) \times Z\)**:
   - **\(X \cap Y\)** represents the intersection of sets \(X\) and \(Y\), which is the portion that belongs to both sets \(X\) and \(Y\).
   - When intersected with \(Z\), you are looking for the part that belongs to both \((X \cap Y)\) and \(Z\).

2. **\((X \cup Y) \times (Z - Y)\)**:
   - **\(X \cup Y\)** represents the union of \(X\) and \(Y\), i.e., all elements that belong to either set \(X\) or set \(Y\).
   - **\(Z - Y\)** represents the difference, i.e., elements in \(Z\) but not in \(Y\).
   - The final result is the intersection of the union of \(X\) and \(Y\) with the set \(Z\) excluding \(Y\).

Would you like a detailed step-by-step explanation of these operations? Here are some related questions to explore the concepts further:

1. What is the difference between union and intersection in set theory?
2. How do you visualize set operations on a Venn diagram?
3. What does the difference operation \((Z - Y)\) represent in a Venn diagram?
4. How can these set expressions be applied in real-life scenarios?
5. How do Cartesian products relate to set operations?

**Tip:** Always remember that intersections focus on common elements between sets, while unions combine all elements from the sets involved.

By using Venn diagram which represents the sets X, Y, Z, find: 1) (X ∩ Y) × Z, 2) (X ∪ Y) × (Z - Y)

Understand how to evaluate set operations using Venn diagrams for three sets X, Y, and Z. Learn to solve expressions like (X ∩ Y) × Z and (X ∪ Y) × (Z - Y) through visual representation and basic set theory.

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