1. Draw Venn diagrams showing 
a) A U B 𝐶 A U C but B ⊈C
b) A ∩ B ∁ A ∩ C but B ⊈C
c) A U B = A U C but B ≠C
d) A ∩ B = A ∩ C but B ≠C

### a) \( A \cup B \subset A \cup C \) but \( B \not\subset C \)
In this scenario, the Venn diagram should show that everything in \( A \cup B \) is contained within \( A \cup C \), but some elements of \( B \) are not in \( C \). This means \( B \) extends outside \( C \), but the union \( A \cup B \) must still be within \( A \cup C \).

**Key details:**
- \( A \) is shared by both unions.
- \( B \) contains elements not in \( C \), but \( A \cup B \subset A \cup C \).

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### b) \( A \cap B \subset A \cap C \) but \( B \not\subset C \)
Here, the intersection of \( A \) and \( B \) is a subset of the intersection of \( A \) and \( C \), but again, some part of \( B \) lies outside \( C \).

**Key details:**
- \( A \cap B \) is fully contained within \( A \cap C \).
- \( B \) still has some elements that are outside \( C \), but they don't affect the intersection.

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### c) \( A \cup B = A \cup C \) but \( B \neq C \)
This shows that although \( A \cup B \) and \( A \cup C \) are the same, \( B \) and \( C \) are not equal. In other words, \( B \) and \( C \) may have different elements, but their union with \( A \) produces the same set.

**Key details:**
- Both unions are identical.
- However, \( B \) and \( C \) differ, meaning they have distinct elements.

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### d) \( A \cap B = A \cap C \) but \( B \neq C \)
In this case, the intersections of \( A \) with \( B \) and \( C \) are the same, but \( B \) and \( C \) are not identical. So \( A \cap B \) and \( A \cap C \) form the same region in the Venn diagram, yet \( B \) and \( C \) are different sets.

**Key details:**
- Intersections are identical.
- \( B \) and \( C \) are not equal.

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Would you like detailed visual diagrams for these?

Draw Venn diagrams showing: a) A U B ⊆ A U C but B ⊈ C, b) A ∩ B ⊆ A ∩ C but B ⊈ C, c) A U B = A U C but B ≠ C, d) A ∩ B = A ∩ C but B ≠ C

Explore the concepts of set union, intersection, and subset relations through Venn diagrams. This problem examines scenarios where A U B ⊆ A U C but B ⊈ C, and more. Suitable for students in Grades 9-12.

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