Question

the universal set is the set of rational numbers. s is the set of integers. which represents ( s^c )? \\( \circ \\) \\( \\{x|x \text{ is a real number}\\}\\) \\( \circ \\) \\( \\{x|x \text{ is a rational number}\\}\\) \\( \circ \\) \\( \\{x|x \text{ is a rational positive number}\\}\\) \\( \circ \\) \\( \\{x|x \text{ is a rational non - integer}\\}\\)

Explanation:

Step1: Recall the definition of complement

The complement of a set \( S \) (denoted \( S^c \)) with respect to a universal set \( U \) is the set of all elements in \( U \) that are not in \( S \). Here, the universal set \( U \) is the set of rational numbers, and \( S \) is the set of integers.

Step2: Determine the elements of \( S^c \)

We need to find all rational numbers that are not integers. So we check each option:

• Option 1: \(\{x|x \text{ is a real number}\}\) includes irrational numbers too, but our universal set is rational numbers, so this is incorrect.
• Option 2: \(\{x|x \text{ is a rational number}\}\) is the universal set, not the complement of integers (since integers are part of rational numbers), so this is incorrect.
• Option 3: \(\{x|x \text{ is a rational positive number}\}\) includes positive integers and positive non - integers, it's not the complement of all integers, so this is incorrect.
• Option 4: \(\{x|x \text{ is a rational non - integer}\}\) is the set of all rational numbers that are not in the set of integers, which is the complement of \( S \) (set of integers) with respect to the universal set of rational numbers.

Answer:

\(\{x|x \text{ is a rational non - integer}\}\) (the fourth option)