Given the subset [tex]\( S = \{x, y, 4, 9, ?\} \)[/tex], we need to determine the most appropriate universal set [tex]\( U \)[/tex] that can describe [tex]\( S \)[/tex]. Let's analyze each option provided for [tex]\( U \)[/tex].
1. Keys on a keyboard:
- The set of keys on a keyboard includes letters, numbers, punctuation marks, and other special symbols.
- While this set does contain all the elements in [tex]\( S \)[/tex], it is a more general and all-encompassing set than necessary for describing [tex]\( S \)[/tex].
2. Letters:
- Letters include only alphabetic characters.
- [tex]\( S \)[/tex] contains 'x' and 'y' which are letters, but it also contains '4', '9', and '?' which are not letters.
- Therefore, this universal set does not fully describe [tex]\( S \)[/tex] since it would exclude some of its elements.
3. Numbers:
- Numbers include digits such as '0' through '9'.
- [tex]\( S \)[/tex] contains '4' and '9', which are numbers.
- Although 'x' and 'y' are not numbers, and '?' is typically not considered a number, 'numbers' could be the most appropriate given the primary focus on numerical digits within the set.
4. Punctuation marks:
- Punctuation marks include symbols like '.', ',', '!', '?', etc.
- [tex]\( S \)[/tex] contains '?', which is a punctuation mark.
- However, 'x', 'y', '4', and '9' are not punctuation marks.
- This universal set does not adequately describe [tex]\( S \)[/tex] since it would exclude the other elements.
Based on the analysis, we see that while 'S' has a mixture of letters, numbers, and a punctuation mark, the most suitable and specific universal set [tex]\( U \)[/tex] that can accurately describe [tex]\( S \)[/tex] is numbers. This is because 'numbers' primarily fits with the numerical elements within [tex]\( S \)[/tex] and can also accept '?' as a symbol representing an unknown or unspecified number.
Thus, the best description for [tex]\( U \)[/tex] is:
[tex]\[ U = \{ \text{numbers} \} \][/tex]
