To determine which description of the universal set [tex]\( U \)[/tex] is most appropriate given that [tex]\( S = \{ x, y, 4, 9, ? \} \)[/tex], we need to analyze the elements within the subset [tex]\( S \)[/tex].
1. The elements of [tex]\( S \)[/tex] are [tex]\( x, y, 4, 9, \text{?} \)[/tex].
- [tex]\( x \)[/tex] is a letter.
- [tex]\( y \)[/tex] is a letter.
- [tex]\( 4 \)[/tex] is a number.
- [tex]\( 9 \)[/tex] is a number.
- The question mark [tex]\( \text{?} \)[/tex] could be interpreted as a placeholder that can vary but let’s try to fit it into potential categories.
Next, we examine the options for [tex]\( U \)[/tex]:
- [tex]\( U = \{\text{keys on a keyboard}\} \)[/tex]
While it's true that [tex]\( x, y, 4, 9, \text{?} \)[/tex] all correspond to keys on a keyboard, this set would include a vast range of other elements like letters, numbers, other punctuation marks, function keys, etc. It is too broad to be a specific description from which [tex]\( S \)[/tex] is derived.
- [tex]\( U = \{\text{letters}\} \)[/tex]
[tex]\( S \)[/tex] contains [tex]\( x \)[/tex] and [tex]\( y \)[/tex] which are letters, but it also contains numbers [tex]\( 4 \)[/tex] and [tex]\( 9 \)[/tex]. Thus, the set of letters only does not fully encapsulate all the elements of [tex]\( S \)[/tex].
- [tex]\( U = \{\text{numbers}\} \)[/tex]
[tex]\( S \)[/tex] includes [tex]\( 4 \)[/tex] and [tex]\( 9 \)[/tex], which are numbers. Since numbers are a specific well-defined category and elements from [tex]\( S \)[/tex] match the idea that they could belong to the set of numbers, this fits well. Letters and numbers are often categories appearing in such questions, and 4 and 9 certainly fit this subset. Another important point is that numbers can encompass different aspects and can suggest a numerical context making this a likely candidate.
- [tex]\( U = \{\text{punctuation marks}\} \)[/tex]
Similarly to the letters option, [tex]\( S \)[/tex] does not consist solely of punctuation marks. It does have a question mark as a potential element, but not the rest. Therefore, this option is also not suitable.
Based on this analysis, the best description of [tex]\( U \)[/tex] that aligns with the elements in [tex]\( S \)[/tex] is:
[tex]\[ U = \{\text{numbers}\} \][/tex]
Hence, the most appropriate description for the universal set [tex]\( U \)[/tex] is the set of all numbers:
[tex]\[ \boxed{3} \][/tex]
