On the Venn diagrams shade the regions (a) A' ∩ C, (b) (A ∪ C) ∩ B.

Understand the Problem

The question is asking to shade specific regions in Venn diagrams based on set operations involving the sets A, B, and C. In part (a), it requires shading the area defined by A' ∩ C, while in part (b), it asks for shading the area defined by (A ∪ C) ∩ B.

Answer

For part (a), shade the region in $C$ not in $A$. For part (b), shade the region in $B$ that overlaps with $A \cup C$.

Steps to Solve

Understanding $A' \cap C$

To find the region defined by $A' \cap C$, we need to identify the area outside of set $A$, represented as $A'$, and the area within set $C$.

Shading the area for part (a)

To shade $A' \cap C$, shade the section of circle $C$ that does not overlap with circle $A$. This means shading the part of circle $C$ that lies outside of circle $A$.

Understanding $(A \cup C) \cap B$

For part (b), we need to first determine the area represented by $A \cup C$, which includes all elements in either set $A$ or set $C$.

Shading the area for part (b)

After identifying $A \cup C$, we intersect this area with set $B$. Shade the region that is part of both the union of sets $A$ and $C$, and also included in set $B$.

The shaded area for part (a) is the region in set $C$ that does not intersect with set $A$. For part (b), the shaded area is the part of set $B$ that intersects with the union of sets $A$ and $C$.

More Information

In Venn diagrams, the operations on sets visualize relationships between different groups. $A'$ represents the complement of $A$, while the union $A \cup C$ includes all elements from both $A$ and $C$.

Tips

Confusing $A'$ with $A$; remember that $A'$ represents everything outside of set $A$.
Not shading the correct intersections or unions; carefully analyze which areas need to be shaded based on the given operations.

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