[tex]$S$[/tex] is a subset within a universal set, [tex]$U$[/tex].

If [tex]$S = \{x, y, 4, 9, 7\}$[/tex], which could describe [tex]$U$[/tex]?

A. [tex]$U = \{$[/tex] keys on a keyboard [tex]$\}$[/tex]
B. [tex]$U = \{$[/tex] letters [tex]$\}$[/tex]
C. [tex]$U = \{$[/tex] numbers [tex]$\}$[/tex]
D. [tex]$U = \{$[/tex] punctuation marks [tex]$\}$[/tex]



Answer :

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].