To determine the appropriate universal set [tex]\( U \)[/tex] for the subset [tex]\( S = \{ x, y, 4, 9, 7 \} \)[/tex], we need to examine the elements within [tex]\( S \)[/tex] carefully and identify which universal set [tex]\( U \)[/tex] can contain all these elements.
1. Consider the possibility that [tex]\( U \)[/tex] is the set of keys on a keyboard:
- The set of keys on a keyboard includes letters, numbers, punctuation marks, and other symbols.
- [tex]\( S \)[/tex] contains [tex]\( x \)[/tex] and [tex]\( y \)[/tex] (which are keyboard keys) and [tex]\( 4, 9, \)[/tex] and [tex]\( 7 \)[/tex] (which are also keyboard keys).
- Therefore, all elements of [tex]\( S \)[/tex] could indeed belong to the set of keys on a keyboard.
2. Consider the possibility that [tex]\( U \)[/tex] is the set of letters:
- The set of letters includes alphabetic characters like [tex]\( x \)[/tex] and [tex]\( y \)[/tex].
- However, [tex]\( S \)[/tex] contains [tex]\( 4, 9, \)[/tex] and [tex]\( 7 \)[/tex], which are not letters.
- Therefore, [tex]\( U \)[/tex] cannot be the set of letters because not all elements in [tex]\( S \)[/tex] are letters.
3. Consider the possibility that [tex]\( U \)[/tex] is the set of numbers:
- The set of numbers includes numeric values such as [tex]\( 4, 9, \)[/tex] and [tex]\( 7 \)[/tex].
- However, [tex]\( S \)[/tex] contains [tex]\( x \)[/tex] and [tex]\( y \)[/tex], which are not numbers.
- This means [tex]\( U \)[/tex] cannot be the set of numbers because not all elements in [tex]\( S \)[/tex] are numbers.
4. Consider the possibility that [tex]\( U \)[/tex] is the set of punctuation marks:
- The set of punctuation marks includes symbols like commas, periods, exclamations, etc.
- [tex]\( S \)[/tex] contains [tex]\( x \)[/tex], [tex]\( y \)[/tex], [tex]\( 4, 9, \)[/tex] and [tex]\( 7 \)[/tex], none of which are punctuation marks.
- Therefore, [tex]\( U \)[/tex] cannot be the set of punctuation marks since none of the elements in [tex]\( S \)[/tex] are punctuation marks.
Based on the analysis above:
- The only universal set [tex]\( U \)[/tex] that appropriately describes the subset [tex]\( S = \{ x, y, 4, 9, 7 \} \)[/tex] is:
[tex]\[
U = \{ \text{keys on a keyboard} \}
\][/tex]
Thus, the correct description of the universal set [tex]\( U \)[/tex] is the set of keys on a keyboard. Therefore, the answer is [tex]\( 3 \)[/tex].
