Let $$ A= $$ \{apple, prune, cantaloupe, pear\}, $$ B= $$ (peach, lemon\}, and $$ C= $$ \{peach, kiwi, berry\}. (a) Find (i) $$ n(A \cup B) $$, (ii) $$ n(A \cup C) $$, and (iii) $$ n(B \cup C) $$. (b) In which case is the number of elements in the union not the sum of the number of elements in the individual sets? (a) (i) $$ n(A \cup B)=6 $$ (ii) $$ n(A \cup C)=7 $$ (iii) $$ n(B \cup C)=4 $$ (b) In which case is the number of elements in the union not the sum of the number of elements in the individual sets? (iii) (i) (ii) (i) AInd (iii) (i) and (ii) (ii) and (iii) (i). (ii), and (iii) None of the cases

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UpStudy AI · Accepted Answer

Let's analyze the given sets and solve each part step-by-step.

**Given Sets:**
- $$ A = \{ \text{apple}, \text{prune}, \text{cantaloupe}, \text{pear} \} $$ → $$ |A| = 4 $$
- $$ B = \{ \text{peach}, \text{lemon} \} $$ → $$ |B| = 2 $$
- $$ C = \{ \text{peach}, \text{kiwi}, \text{berry} \} $$ → $$ |C| = 3 $$

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### **Part (a): Calculate the Number of Elements in Unions**

1. **$$ n(A \cup B) $$:**
   - **Union of A and B:**
     $$
     A \cup B = \{ \text{apple}, \text{prune}, \text{cantaloupe}, \text{pear}, \text{peach}, \text{lemon} \}
     $$
   - **Number of Elements:**
     $$
     n(A \cup B) = 6
     $$

2. **$$ n(A \cup C) $$:**
   - **Union of A and C:**
     $$
     A \cup C = \{ \text{apple}, \text{prune}, \text{cantaloupe}, \text{pear}, \text{peach}, \text{kiwi}, \text{berry} \}
     $$
   - **Number of Elements:**
     $$
     n(A \cup C) = 7
     $$

3. **$$ n(B \cup C) $$:**
   - **Union of B and C:**
     $$
     B \cup C = \{ \text{peach}, \text{lemon}, \text{kiwi}, \text{berry} \}
     $$
   - **Number of Elements:**
     $$
     n(B \cup C) = 4
     $$

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### **Part (b): When is the Union Not the Sum of Individual Sets?**

The number of elements in the union of two sets is **not** simply the sum of the number of elements in each set **if and only if** the sets have at least one element in common. This is because the overlapping elements are counted twice when simply adding the sizes.

**Analyzing Each Case:**

1. **$$ A \cup B $$:**
   - **Common Elements:** None
   - **Sum of Sizes:** $$ |A| + |B| = 4 + 2 = 6 $$
   - **Union Size:** $$ n(A \cup B) = 6 $$
   - **Conclusion:** $$ n(A \cup B) = |A| + |B| $$

2. **$$ A \cup C $$:**
   - **Common Elements:** None
   - **Sum of Sizes:** $$ |A| + |C| = 4 + 3 = 7 $$
   - **Union Size:** $$ n(A \cup C) = 7 $$
   - **Conclusion:** $$ n(A \cup C) = |A| + |C| $$

3. **$$ B \cup C $$:**
   - **Common Elements:** $$ \text{peach} $$
   - **Sum of Sizes:** $$ |B| + |C| = 2 + 3 = 5 $$
   - **Union Size:** $$ n(B \cup C) = 4 $$
   - **Conclusion:** $$ n(B \cup C) < |B| + |C| $$

**Therefore, the union does not equal the sum of the individual set sizes only in case (iii): $$ B \cup C $$.**

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**Final Answers:**

- **Part (a):**
  - (i) $$ n(A \cup B) = 6 $$
  - (ii) $$ n(A \cup C) = 7 $$
  - (iii) $$ n(B \cup C) = 4 $$

- **Part (b):**
  - **Only in case (iii): $$ B \cup C $$**
- **Part (a):** - (i) $$ n(A \cup B) = 6 $$ - (ii) $$ n(A \cup C) = 7 $$ - (iii) $$ n(B \cup C) = 4 $$ - **Part (b):** - The union does not equal the sum of the individual set sizes only in case (iii): $$ B \cup C $$.

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