Work the following problem, similar to number 19 from
section 1.1 of your text. Let U = {1, 2, 3, 4, b, x, y} be a
universal set with subsets * = {1, 2, 4, x}, Y = {1, 3, x},
and A = {1,2, 3, y}. Another subset of U is given by
S = 12, 3, 4, 6, y}.
Which of the following sets are needed to express the set S?

To determine which sets are needed to express the set \( S = \{1, 2, 3, 4, 6, y\} \), we should compare the elements of \( S \) with the elements in the given subsets \( X \), \( Y \), and \( A \) of the universal set \( U \).

### Step 1: List out the given sets.
- **Universal set \( U \):** \{1, 2, 3, 4, b, x, y\}
- **Subset \( X \):** \{1, 2, 4, x\}
- **Subset \( Y \):** \{1, 3, x\}
- **Subset \( A \):** \{1, 2, 3, y\}

### Step 2: List out the elements of \( S \).
- \( S = \{1, 2, 3, 4, 6, y\} \)

### Step 3: Compare \( S \) with the subsets \( X \), \( Y \), and \( A \).

- **Elements in \( S \) but not in any subset:**
  - \( 6 \) is not in \( U \), hence it cannot be in \( X \), \( Y \), or \( A \).
  - However, for \( S = \{1, 2, 3, 4, 6, y\} \), \( S \) needs to be compared with elements of the universal set and the given subsets.

Let's check each element:
- \( 1 \) is in \( X \), \( Y \), and \( A \).
- \( 2 \) is in \( X \) and \( A \).
- \( 3 \) is in \( Y \) and \( A \).
- \( 4 \) is in \( X \).
- \( y \) is in \( A \).
- \( 6 \) is not in any of the subsets given but must be in \( S \).

### Step 4: Express \( S \) using unions and intersections of the subsets.

Since \( S \) contains elements from each of the subsets \( X \), \( Y \), and \( A \), but \( S \) also includes the element \( 6 \), which is not in any subset, we realize \( S \) cannot be expressed purely as a combination of these subsets.

### Conclusion:
\( S \) cannot be expressed exactly using the given subsets \( X \), \( Y \), and \( A \), because \( S \) includes the element \( 6 \), which is not in the universal set \( U \) and hence not in any of the subsets.

### However, if we ignore the non-existent element \( 6 \), the set \( S \) could be approximately expressed as the union of the given subsets:

\[ S \approx X \cup Y \cup A \]

But since \( 6 \) is not in \( U \), strictly speaking, \( S \) cannot be constructed entirely from the given subsets.

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Do you want any further details or have any questions?

**Relative Questions:**
1. How can we express the union of two sets \( X \) and \( Y \)?
2. What is the intersection of the sets \( X \) and \( A \) from the problem?
3. Can you define what the universal set \( U \) represents in set theory?
4. How do we determine if an element belongs to a given set?
5. Can a set be expressed if it includes an element not present in the universal set?

**Tip:** When working with set operations, always ensure all elements belong to the universal set to make valid expressions and comparisons.

Explore how to express set S = {1, 2, 3, 4, 6, y} using subsets X = {1, 2, 4, x}, Y = {1, 3, x}, and A = {1, 2, 3, y}. Understand the challenges posed by including elements not present in the universal set U. Ideal for advanced high school students studying set theory.

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