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?