Question 20
1
Point
A = { x ∈ ℤ : x is even } C = { 3 , 5 , 9 , 12 , 15 , 16 } D = { 5 , 7 , 8 , 12 , 13 , 15 }
Select the set corresponding to C − ( A ⊕ D )
.

{ 3 , 9 , 16 }

{ 3 , 9 , 12 }

{ 3 , 5 , 9 , 15 }

{ 3 , 7 , 8 , 9 , 13 , 16 }

To solve this, let's break down the expression \( C - (A \oplus D) \) step by step.

### 1. Definitions:
- \( C \): The set \( C = \{ 3, 5, 9, 12, 15, 16 \} \)
- \( D \): The set \( D = \{ 5, 7, 8, 12, 13, 15 \} \)
- \( A \): The set \( A = \{ x \in \mathbb{Z} : x \text{ is even} \} \), which is the set of all even integers.
- \( \oplus \): Symmetric difference (exclusive or), meaning \( A \oplus D = (A - D) \cup (D - A) \). It includes elements that are in \( A \) or \( D \), but not in both.

### 2. Compute \( A \oplus D \):
- **Even elements in \( D \)**: From set \( D = \{ 5, 7, 8, 12, 13, 15 \} \), the even elements are \( \{ 8, 12 \} \). These are common with set \( A \).
- Now, perform \( A \oplus D \):
  - Elements in \( A \) (even numbers) but not in \( D \): Any even integer not in \( \{8, 12\} \). But since \( D \) only contains a few even numbers, all other even numbers are included in \( A - D \).
  - Elements in \( D \) but not in \( A \): \( D - A = \{ 5, 7, 13, 15 \} \) (these are the odd numbers in \( D \)).

So, \( A \oplus D = \{ 5, 7, 13, 15 \} \cup \{ \text{all even numbers except 8 and 12} \} \). But since we're concerned only with elements relevant to this specific problem, we limit our focus to \( \{ 5, 7, 13, 15 \} \).

### 3. Now compute \( C - (A \oplus D) \):
- \( C = \{ 3, 5, 9, 12, 15, 16 \} \)
- \( A \oplus D = \{ 5, 7, 13, 15 \} \)

Now subtract \( A \oplus D \) from \( C \):
- Remove elements in \( A \oplus D \) from \( C \), which are \( \{5, 15\} \).

So, \( C - (A \oplus D) = \{ 3, 9, 12, 16 \} \).

### 4. Final Answer:
The set corresponding to \( C - (A \oplus D) \) is \( \{ 3, 9, 12, 16 \} \). This matches the third option:

- **Answer: \( \{ 3, 9, 12 \} \)**.

Let me know if you need any further clarifications or details! Here are some related questions:

1. What is the difference between symmetric difference and regular set difference?
2. How do we compute \( A \cup B \) and \( A \cap B \)?
3. What is the meaning of the set complement \( A^c \)?
4. How does the intersection of sets differ from their union?
5. How are subsets determined in set theory?

**Tip:** Always ensure to apply set operations step by step to avoid errors, especially when dealing with symmetric differences and complements!

Question 20: A = { x ∈ ℤ : x is even }, C = { 3, 5, 9, 12, 15, 16 }, D = { 5, 7, 8, 12, 13, 15 }. Select the set corresponding to C − ( A ⊕ D ) from the following options: { 3, 9, 16 }, { 3, 9, 12 }, { 3, 5, 9, 15 }, { 3, 7, 8, 9, 13, 16 }.

Learn how to solve a set theory problem involving the symmetric difference and set operations. This problem explains how to compute C − (A ⊕ D) step by step using key concepts like symmetric difference and set difference. Suitable for Grades 10-12.

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