Questions: Set Operations and Venn Diagrams with Use the Venn diagram shown to the right to answer the following questions. a. Which regions are represented by A ∩(B ∪ C) ? b. Which regions are represented by (A ∩ B) ∪(A ∩ C) ? c. Based on parts (a) and (b), are A ∩(B ∪ C) and (A ∩ B) ∪(A ∩ C) equal for all sets A and B ? a. Select the regions that represent A ∩(B ∪ C).

Set Operations and Venn Diagrams with Use the Venn diagram shown to the right to answer the following questions. a. Which regions are represented by A ∩(B ∪ C) ? b. Which regions are represented by (A ∩ B) ∪(A ∩ C) ? c. Based on parts (a) and (b), are A ∩(B ∪ C) and (A ∩ B) ∪(A ∩ C) equal for all sets A and B ? a. Select the regions that represent A ∩(B ∪ C).
Transcript text: et Operations and Venn Diagrams with Use the Venn diagram shown to the right to answer the following questions. a. Which regions are represented by $A \cap(B \cup C)$ ? b. Which regions are represented by $(A \cap B) \cup(A \cap C)$ ? c. Based on parts (a) and (b), are $A \cap(B \cup C)$ and $(A \cap B) \cup(A \cap C)$ equal for all sets $A$ and $B$ ? a. Select the regions that represent $A \cap(B \cup C)$.
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Solution

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Solution Steps

Step 1: Identify regions for \( A \cap (B \cup C) \)
  • \( B \cup C \) includes all regions in sets \( B \) and \( C \): II, III, V, VI, and VII.
  • \( A \cap (B \cup C) \) includes regions in both \( A \) and \( B \cup C \): II and V.
Step 2: Identify regions for \( (A \cap B) \cup (A \cap C) \)
  • \( A \cap B \) includes regions in both \( A \) and \( B \): II and V.
  • \( A \cap C \) includes regions in both \( A \) and \( C \): V and VI.
  • \( (A \cap B) \cup (A \cap C) \) includes regions in either \( A \cap B \) or \( A \cap C \): II, V, and VI.
Step 3: Compare \( A \cap (B \cup C) \) and \( (A \cap B) \cup (A \cap C) \)
  • \( A \cap (B \cup C) \) includes regions: II and V.
  • \( (A \cap B) \cup (A \cap C) \) includes regions: II, V, and VI.
  • They are not equal for all sets \( A \) and \( B \).

Final Answer

  1. \( A \cap (B \cup C) \) includes regions: II and V.
  2. \( (A \cap B) \cup (A \cap C) \) includes regions: II, V, and VI.
  3. \( A \cap (B \cup C) \) and \( (A \cap B) \cup (A \cap C) \) are not equal for all sets \( A \) and \( B \).
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