Give a Venn Diagram for each of the following assume A, B, C ⊆ U: (A ∪ B ∪ C) ∩ ((A ∩ B) ∪ (A ∩ C) ∪ (B ∩ C))
Understand the Problem
The question is asking for a Venn Diagram to illustrate the set operations defined using the complements and intersections of the sets A, B, and C. Specifically, it involves the union of the complements and their intersection with the union of intersections of the sets.
Answer
$$(A^c \cup B^c \cup C^c) \cap ((A \cap B) \cup (A \cap C) \cup (B \cap C))$$
Answer for screen readers
The final expression is given by: $$(A^c \cup B^c \cup C^c) \cap ((A \cap B) \cup (A \cap C) \cup (B \cap C))$$
Steps to Solve
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Identify the Sets and Complements We have three sets: ( A ), ( B ), and ( C ). Their complements are ( A^c ), ( B^c ), and ( C^c ).
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Determine the Union of Complements We first find the union of the complements: $$ A^c \cup B^c \cup C^c $$ This represents all elements that are not in any of the sets ( A ), ( B ), or ( C ).
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Find the Intersections of the Sets Next, find the intersections of the sets: $$ A \cap B, \quad A \cap C, \quad B \cap C $$ These represent all elements that are common between each pair of sets.
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Combine Intersections into a Union Now, combine these intersections into one expression: $$ (A \cap B) \cup (A \cap C) \cup (B \cap C) $$ This represents all elements that are in at least two of the sets.
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Final Intersection of the Two Results We need to intersect the union of complements with the union of intersections: $$ (A^c \cup B^c \cup C^c) \cap ((A \cap B) \cup (A \cap C) \cup (B \cap C)) $$ This gives us the final set of elements that are in the complements and also among the pairwise intersections.
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Illustrate with a Venn Diagram To represent this visually, draw three overlapping circles for ( A ), ( B ), and ( C ). Shade the areas that correspond to ( (A^c \cup B^c \cup C^c) ) that also fall in the shaded areas of the pairwise intersections.
The final expression is given by: $$(A^c \cup B^c \cup C^c) \cap ((A \cap B) \cup (A \cap C) \cup (B \cap C))$$
More Information
This expression highlights elements that are outside all sets ( A ), ( B ), and ( C ) while still being part of at least two of the sets. The Venn Diagram helps visualize overlapping relationships among the sets and their complements.
Tips
- Misunderstanding complements: Ensure you grasp what ( A^c ), ( B^c ), and ( C^c ) represent (elements outside those sets).
- Confusing union and intersection: Remember that union captures all unique elements, while intersection only includes those shared among sets.
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