Final answer:
The correct set that includes rational numbers but not natural numbers is a) (-5, -4, 4, 5), because it contains negative integers, which are rational but not considered natural numbers, and does not include any strictly positive integers without fractions.
Step-by-step explanation:
The question involves identifying which set includes rational numbers but not natural numbers. Rational numbers are numbers that can be expressed as a fraction with an integer numerator and a non-zero integer denominator, which includes fractions, integers (both positive and negative), and the number zero. Natural numbers are the set of positive integers starting from 1. Therefore, we are looking for a set that includes fractions or negative numbers (which are rational but not natural), but no natural numbers.
Looking at the options:
- a) (-5, -4, 4, 5) includes negative integers, which are rational but not natural numbers.
- b) (5, 7, 8) includes only natural numbers, which are also rational but doesn't fit the criteria as it includes natural numbers.
- c) (0, 1, 2, 3) includes natural numbers and the number 0, which is rational but not a natural number. However, since natural numbers are also included, this option is not correct.
- d) (5, frac(1)/(8), frac(5)/(18)) includes the fraction 1/8 and 5/18, which are rational numbers but it also includes the natural number 5.
Hence, the correct option is a) (-5, -4, 4, 5) as it includes negative integers and a positive integer which are all rational, but it does not include any natural numbers (since natural numbers are strictly positive).