Which of the following has no solution? \[ \{x \mid x \leq 0\} \text { and }[x \mid x \geq 0\} \] \( \{x \mid x<0\} \) and \( \{x \mid x>0\} \) \( [x \mid x \leq 0\} \) or \( [x \mid x \geq 0\} \)

To determine which of the given sets has no solution, we need to analyze each pair of sets. 1. **First Pair: \( \{x \mid x \leq 0\} \) and \( \{x \mid x \geq 0\} \)** - The first set includes all values of \( x \) that are less than or equal to 0. - The second set includes all values of \( x \) that are greater than or equal to 0. - The intersection of these two sets is \( x = 0 \), which means there is a solution (specifically, \( x = 0 \)). 2. **Second Pair: \( \{x \mid x < 0\} \) and \( \{x \mid x > 0\} \)** - The first set includes all values of \( x \) that are strictly less than 0. - The second set includes all values of \( x \) that are strictly greater than 0. - There are no values of \( x \) that can satisfy both conditions simultaneously, so this pair has no solution. 3. **Third Pair: \( \{x \mid x \leq 0\} \) and \( \{x \mid x \geq 0\} \)** - This is similar to the first pair. The first set includes all values of \( x \) that are less than or equal to 0, and the second set includes all values of \( x \) that are greater than or equal to 0. - The intersection is again \( x = 0 \), which means there is a solution. ### Conclusion: The pair that has no solution is: \[ \{x \mid x < 0\} \text{ and } \{x \mid x > 0\} \]