To identify which set includes rational numbers but not natural numbers, let's first clarify the definitions:
- Natural numbers are the positive integers starting from 1 onwards: {1, 2, 3, 4, ...}.
- Rational numbers are numbers that can be expressed as a fraction \( \frac{a}{b} \), where \( a \) and \( b \) are integers and \( b \neq 0 \). This category includes positive and negative fractions, integers, and zero.
Now, let's analyze each set:
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Set 1: Negative 6 and two-thirds, negative 2, \( \frac{5}{11} \), \( \frac{7}{8} \)
- Contains: -6 (integer), -2 (integer), \( \frac{5}{11} \) (rational), \( \frac{7}{8} \) (rational).
- Does not contain natural numbers. Contains rational numbers.
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Set 2: Negative 5, negative 4, 4, 5
- Contains: -5 (integer), -4 (integer), 4 (natural number), 5 (natural number).
- Contains natural numbers.
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Set 3: Three-fourths, 4 and \( \frac{5}{6} \), \( \frac{18}{5} \), 8
- Contains: \( \frac{3}{4} \) (rational), \( \frac{5}{6} \) (rational), \( \frac{18}{5} \) (rational), 4 (natural number), 8 (natural number).
- Contains natural numbers.
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Set 4: 0, 1, 2, 3
- Contains: 0 (rational), 1 (natural number), 2 (natural number), 3 (natural number).
- Contains natural numbers.
Now, we can conclude:
The only set that includes rational numbers but does not include natural numbers is Set 1 (Negative 6 and two-thirds, negative 2, \( \frac{5}{11} \), \( \frac{7}{8} \)).