To find \( S^c \), the complement of \( S \) in the universal set of rational numbers, we need to consider the elements that are in the universal set (the set of rational numbers) but not in the set of integers \( S \).
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The universal set is the set of rational numbers, which can be defined as: \[ { x \mid x \text{ is a rational number} } \]
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The set of integers \( S \) is: \[ S = { x \mid x \text{ is an integer} } \]
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The complement \( S^c \) would therefore include all rational numbers that are not integers. This can be defined as: \[ S^c = { x \mid x \text{ is a rational number} } \setminus { x \mid x \text{ is an integer} } \] or simply: \[ S^c = { x \mid x \text{ is a rational number and not an integer} } \]
Among the options given, the one that correctly represents \( S^c \) is:
- {x | x is a rational non-integer}
Thus, the correct representation of \( S^c \) is: {x | x is a rational non-integer}.
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