Answer :
To determine the best description of the universal set \( U \) that contains the subset \( S = \{x, y, 4, 9\} \), we need to analyze the elements in \( S \).
The elements of \( S \) are:
- \( x \): which is a letter.
- \( y \): which is a letter.
- \( 4 \): which is a number.
- \( 9 \): which is a number.
Since \( S \) contains both letters and numbers, we can rule out any universal sets that do not accommodate both types of elements.
Let's analyze the options for \( U \) one by one:
1. \( U = \{ \text{keys on a keyboard} \} \):
- This set includes letters, numbers, punctuation marks, and other special keys.
- This set is too broad because it includes more than just letters and numbers.
2. \( U = \{ \text{letters} \} \):
- This set includes only letters, so it cannot describe \( S \) completely because \( S \) also contains numbers.
3. \( U = \{ \text{numbers} \} \):
- This set includes only numbers, so it cannot describe \( S \) completely because \( S \) also contains letters.
4. \( U = \{ \text{punctuation marks} \} \):
- This set includes only punctuation marks and cannot describe \( S \) at all since \( S \) contains letters and numbers but no punctuation marks.
After evaluating all the options, none of them perfectly fit both letters and numbers simultaneously. Therefore, the optimal choice would be the one that best encompasses the diverse nature of \( S \). Despite not being a perfect match, the option describing numbers is the closest fit to comprise some of the subset \( S \), suggesting that options describing only letters or only punctuation marks are less compatible.
Thus, considering the general nature of the elements involved in the set \( S \), the answer is:
[tex]\[ U = \{ \text{numbers} \} \][/tex]
So, the best description for \( U \) is that \( U \) includes numbers, which leads us to the numerical result:
[tex]\[ \text{The best description is } 3 \][/tex]
The elements of \( S \) are:
- \( x \): which is a letter.
- \( y \): which is a letter.
- \( 4 \): which is a number.
- \( 9 \): which is a number.
Since \( S \) contains both letters and numbers, we can rule out any universal sets that do not accommodate both types of elements.
Let's analyze the options for \( U \) one by one:
1. \( U = \{ \text{keys on a keyboard} \} \):
- This set includes letters, numbers, punctuation marks, and other special keys.
- This set is too broad because it includes more than just letters and numbers.
2. \( U = \{ \text{letters} \} \):
- This set includes only letters, so it cannot describe \( S \) completely because \( S \) also contains numbers.
3. \( U = \{ \text{numbers} \} \):
- This set includes only numbers, so it cannot describe \( S \) completely because \( S \) also contains letters.
4. \( U = \{ \text{punctuation marks} \} \):
- This set includes only punctuation marks and cannot describe \( S \) at all since \( S \) contains letters and numbers but no punctuation marks.
After evaluating all the options, none of them perfectly fit both letters and numbers simultaneously. Therefore, the optimal choice would be the one that best encompasses the diverse nature of \( S \). Despite not being a perfect match, the option describing numbers is the closest fit to comprise some of the subset \( S \), suggesting that options describing only letters or only punctuation marks are less compatible.
Thus, considering the general nature of the elements involved in the set \( S \), the answer is:
[tex]\[ U = \{ \text{numbers} \} \][/tex]
So, the best description for \( U \) is that \( U \) includes numbers, which leads us to the numerical result:
[tex]\[ \text{The best description is } 3 \][/tex]
