To identify the set that includes rational numbers but not natural numbers, we first need to understand the definitions:
- Natural numbers are the set of positive integers starting from 1 (i.e., 1, 2, 3, ...).
- Rational numbers include all numbers that can be expressed as a fraction of two integers, which means they can be positive, negative, or zero.
Now, let's evaluate each of the provided sets:
-
Set 1: -1, -6, 3, 8
- This set includes negative numbers and the positive natural number 3 and 8. Hence, it contains natural numbers.
-
Set 2: -1, -6, 3, 86
- Similar to the first set, it includes negative numbers and positive natural numbers (3 and 86). Hence, it contains natural numbers.
-
Set 3: 23, -5, 45, 276
- This set contains the natural number 23 and also includes a negative number (-5). Hence, it contains natural numbers.
-
Set 4: -456, 187, 214
- This set only includes negative and positive rational numbers, but does not include any natural numbers (there are no positive integers starting from 1).
-
Set 5: -456, 187, 20, 1, 2, 3
- This set includes the natural numbers 1, 2, and 3. Hence, it contains natural numbers.
Thus, the only set that includes rational numbers but not natural numbers is Set 4: -456, 187, 214.