The universal set is the set of rational numbers. S is the set of integers.

Question

Which represents Sc?

{x|x is a real number}
{x|x is a rational number}
{x|x is a rational positive number}
{x|x is a rational non-integer}

Answer 1

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 $S$.

1. The universal set is the set of rational numbers, which can be defined as: $$x\mid x\text{ is a rational number}$$

2. The set of integers $S$ is: $$S=x\mid x\text{ is an integer}$$

3. 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}.