To determine which option best describes the universal set [tex]\( U \)[/tex], given that [tex]\( S = \{ x, y, 4, 9 \} \)[/tex] is a subset of [tex]\( U \)[/tex], let's analyze each element of [tex]\( S \)[/tex] and the provided options for [tex]\( U \)[/tex].
First, look at the elements in set [tex]\( S \)[/tex]:
- [tex]\( x \)[/tex] and [tex]\( y \)[/tex] are alphabetical characters (letters).
- [tex]\( 4 \)[/tex] and [tex]\( 9 \)[/tex] are numerical characters (numbers).
Next, we evaluate the given descriptions for [tex]\( U \)[/tex]:
1. Keys on a keyboard:
- This description includes a broad range of characters such as letters, numbers, punctuation marks, and other symbols.
- Since this is too broad, it doesn't specifically match the elements of [tex]\( S \)[/tex].
2. Letters:
- This description matches the alphabetical characters in [tex]\( S \)[/tex] (i.e., [tex]\( x \)[/tex] and [tex]\( y \)[/tex]).
- However, it does not accommodate the numerical characters [tex]\( 4 \)[/tex] and [tex]\( 9 \)[/tex].
3. Numbers:
- This description matches the numerical characters in [tex]\( S \)[/tex] (i.e., [tex]\( 4 \)[/tex] and [tex]\( 9 \)[/tex]).
- However, it does not accommodate the alphabetical characters [tex]\( x \)[/tex] and [tex]\( y \)[/tex].
4. Punctuation marks:
- None of the elements in [tex]\( S \)[/tex] (i.e., [tex]\( x \)[/tex], [tex]\( y \)[/tex], [tex]\( 4 \)[/tex], and [tex]\( 9 \)[/tex]) are punctuation marks.
- Therefore, this description does not describe [tex]\( S \)[/tex] at all.
Based on the analysis, the only valid option that accommodates all the elements of [tex]\( S \)[/tex], taking into account the nature of the elements, is:
Option 2: Letters
