1. **Problem Statement:** We are given a sorted set $S = \{3,7,10,15,23,42,99\}$ and a target value $x = 23$. We want to find the position of $x$ in $S$ using binary search. 2. **Binary Search Formula and Rules:** Binary search works by repeatedly dividing the search interval in half. We start with two pointers: $low = 0$ and $high = n-1$ where $n$ is the number of elements in $S$. We calculate the middle index $mid = \left\lfloor \frac{low + high}{2} \right\rfloor$. If $S[mid] = x$, we found the target. If $S[mid] < x$, we search the right half by setting $low = mid + 1$. If $S[mid] > x$, we search the left half by setting $high = mid - 1$. We repeat until $low > high$ or the target is found. 3. **Step-by-step Execution:** - Initial: $low = 0$, $high = 6$ (since there are 7 elements) - Calculate $mid = \left\lfloor \frac{0 + 6}{2} \right\rfloor = 3$ - Check $S[3] = 15$ which is less than $23$, so set $low = mid + 1 = 4$ - Now $low = 4$, $high = 6$ - Calculate $mid = \left\lfloor \frac{4 + 6}{2} \right\rfloor = 5$ - Check $S[5] = 42$ which is greater than $23$, so set $high = mid - 1 = 4$ - Now $low = 4$, $high = 4$ - Calculate $mid = \left\lfloor \frac{4 + 4}{2} \right\rfloor = 4$ - Check $S[4] = 23$ which equals the target $x$ 4. **Conclusion:** The target value $23$ is found at index $4$ in the sorted set $S$. **Final answer:** $4$ (index of $23$ in $S$)