Give the Venn diagram for the following assume A, B, C ⊆ U: (A ∩ B ∩ C) ∪ (A ∩ Bc ∩ Cc) ∪ (Ac ∩ B ∩ Cc) ∪ (Ac ∩ Bc ∩ C)
Understand the Problem
The question is asking for a Venn diagram representation of the given set operations involving sets A, B, and C. It includes intersections and unions of the sets, which needs to be visually depicted.
Answer
The Venn diagram representation includes shaded areas for all specified intersections and unions.
Answer for screen readers
The Venn diagram represents the union of the shaded areas corresponding to the following regions: $$(A \cap B \cap C) \cup (A \cap B^c \cap C^c) \cup (A^c \cap B \cap C^c) \cup (A^c \cap B^c \cap C)$$
Steps to Solve
- Identify the sets and their operations
We have sets ( A, B, ) and ( C ) and need to analyze the expression $$(A \cap B \cap C) \cup (A \cap B^c \cap C^c) \cup (A^c \cap B \cap C^c) \cup (A^c \cap B^c \cap C)$$ where ( B^c ) and ( C^c ) represent the complements of sets ( B ) and ( C ), respectively.
- Break down the expression into parts
Let's rewrite the components for clarity:
- ( A \cap B \cap C ): Intersection of all three sets ( A, B, ) and ( C )
- ( A \cap B^c \cap C^c ): Elements in ( A ) but outside ( B ) and ( C )
- ( A^c \cap B \cap C^c ): Elements in ( B ) but outside ( A ) and ( C )
- ( A^c \cap B^c \cap C ): Elements in ( C ) but outside ( A ) and ( B )
- Draw the Venn diagram
To create the Venn diagram:
- Draw three overlapping circles representing the sets ( A, B, ) and ( C ).
- Shade the areas corresponding to the components:
- Shade the intersection area for ( A \cap B \cap C ).
- Shade the area only in ( A ) that does not overlap with ( B ) or ( C ) for ( A \cap B^c \cap C^c ).
- Shade the area only in ( B ) that does not overlap with ( A ) or ( C ) for ( A^c \cap B \cap C^c ).
- Shade the area only in ( C ) that does not overlap with ( A ) or ( B ) for ( A^c \cap B^c \cap C ).
- Combine the shaded regions
The final shaded area will illustrate all the conditions specified by the operations in the original expression.
The Venn diagram represents the union of the shaded areas corresponding to the following regions: $$(A \cap B \cap C) \cup (A \cap B^c \cap C^c) \cup (A^c \cap B \cap C^c) \cup (A^c \cap B^c \cap C)$$
More Information
This expression visually shows which elements belong to the specified combinations of the sets ( A, B, ) and ( C ). Understanding Venn diagrams is crucial for visualizing relationships among multiple sets in set theory.
Tips
- Ignoring complements: Sometimes, students forget to include the complement portions in their shading.
- Miscounting intersections: Accurately finding the correct area for intersections can be tricky; double-check the overlaps.
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