To determine the most suitable universal set for performing set operations on the sets [tex]\( A \)[/tex] and [tex]\( B \)[/tex], let's closely examine the elements of each set and compare them to the given sets.
Set [tex]\( A \)[/tex]:
[tex]\[ A = \{-1, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20\} \][/tex]
Set [tex]\( B \)[/tex]:
[tex]\[ B = \left\{0, 3, \frac{10}{3}, 1\right\} \][/tex]
Now, let's analyze both sets in light of the provided options for the universal set:
- Option A: The set of natural numbers:
- Natural numbers include [tex]\(\{1, 2, 3, 4, \ldots\}\)[/tex].
- However, neither [tex]\( -1 \)[/tex] nor [tex]\( 0 \)[/tex] nor fractional numbers (like [tex]\(\frac{10}{3}\)[/tex]) are natural numbers.
- Since [tex]\( -1 \)[/tex] and [tex]\(\frac{10}{3}\)[/tex] are included in sets [tex]\( A \)[/tex] and [tex]\( B \)[/tex], natural numbers are not sufficient as a universal set.
- Option B: The set of integer numbers:
- Integers include [tex]\(\{\ldots, -2, -1, 0, 1, 2, 3, \ldots\}\)[/tex].
- While integers cover most elements, they do not include fractional numbers such as [tex]\(\frac{10}{3}\)[/tex].
- Therefore, the set of integers is not comprehensive enough, given [tex]\(\frac{10}{3}\)[/tex] in set [tex]\( B \)[/tex].
- Option C: The set of irrational numbers:
- Irrational numbers include numbers like [tex]\(\pi\)[/tex] and [tex]\(\sqrt{2}\)[/tex], which cannot be expressed as a ratio of two integers.
- Because none of the elements in sets [tex]\( A \)[/tex] and [tex]\( B \)[/tex] are irrational numbers, irrational numbers cannot serve as the universal set.
- Option D: The set of rational numbers:
- Rational numbers are numbers that can be expressed as a ratio of two integers, which includes integers and fractions.
- The set [tex]\(\{-1, 1, 2, 3, ..., 20\}\)[/tex] in set [tex]\( A \)[/tex] and the set [tex]\(\{0, 3, \frac{10}{3}, 1\}\)[/tex] in set [tex]\( B \)[/tex] are all rational numbers.
- Therefore, the rational numbers set comprehensively covers all elements in both sets [tex]\( A \)[/tex] and [tex]\( B \)[/tex].
Given the analysis, the set of rational numbers is the best choice as the universal set for performing set operations on sets [tex]\( A \)[/tex] and [tex]\( B \)[/tex].
Thus, the answer is:
[tex]\[ \boxed{4} \][/tex]
[tex]\[ \text{D. The set of rational numbers.} \][/tex]
