Q3) If = {2 , 3 , 5 , 7 , 9}, B = {1 , 3 , 5 , 7}, C = {2, 3, 4, 5} , then verify that
(A ∩ B) ∩ C = A ∩ (B ∩ C).


To verify that \((A \cap B) \cap C = A \cap (B \cap C)\), we need to perform the set operations step by step and check both sides of the equation.

Given sets:
- \( A = \{2, 3, 5, 7, 9\} \)
- \( B = \{1, 3, 5, 7\} \)
- \( C = \{2, 3, 4, 5\} \)

First, let's compute \( (A \cap B) \cap C \):

1. **Find \( A \cap B \)**:
\[ A \cap B = \{2, 3, 5, 7, 9\} \cap \{1, 3, 5, 7\} \]
Only the common elements in both sets will be included:
\[ A \cap B = \{3, 5, 7\} \]

2. **Find \( (A \cap B) \cap C \)**:
\[ (A \cap B) \cap C = \{3, 5, 7\} \cap \{2, 3, 4, 5\} \]
Only the common elements in both sets will be included:
\[ (A \cap B) \cap C = \{3, 5\} \]

Next, let's compute \( A \cap (B \cap C) \):

1. **Find \( B \cap C \)**:
\[ B \cap C = \{1, 3, 5, 7\} \cap \{2, 3, 4, 5\} \]
Only the common elements in both sets will be included:
\[ B \cap C = \{3, 5\} \]

2. **Find \( A \cap (B \cap C) \)**:
\[ A \cap (B \cap C) = \{2, 3, 5, 7, 9\} \cap \{3, 5\} \]
Only the common elements in both sets will be included:
\[ A \cap (B \cap C) = \{3, 5\} \]

Since both \((A \cap B) \cap C\) and \(A \cap (B \cap C)\) result in \(\{3, 5\}\), we have verified that:

\[ (A \cap B) \cap C = A \cap (B \cap C) \]

Would you like any further details or have any questions?

Here are 5 questions you might consider next:
1. How do you find the union of two sets?
2. What is the difference between intersection and union of sets?
3. How can you verify the associative property of union?
4. What is the complement of a set?
5. How do you represent sets using Venn diagrams?

**Tip:** When working with set operations, it helps to draw Venn diagrams to visually verify the results.

Explore how to verify the intersection of sets (A ∩ B) ∩ C = A ∩ (B ∩ C) with detailed step-by-step calculations. This problem involves set theory concepts and demonstrates the associative property of set intersection. Suitable for high school students studying set theory.

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