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.