Answer
See explanation
Work Step by Step
We are asked to trace **Algorithm 6.1.1** on the following input:
- \(m = 3,\; n = 3\)
- \(a[1] = u,\; a[2] = v,\; a[3] = w\) → Set \(A = \{u, v, w\}\)
- \(b[1] = w,\; b[2] = u,\; b[3] = v\) → Set \(B = \{w, u, v\}\)
The goal is to test whether \(A \subseteq B\) using the given **algorithm**.
---
## Initial Values
```text
i := 1
answer := "A ⊆ B"
```
Now we enter the outer `while` loop.
---
## 🔁 First iteration (i = 1)
- `a[1] = u`
**Inner loop: searching b[1..3] for `u`:**
- j = 1 → b[1] = w ≠ u → found = "no"
- j = 2 → b[2] = u = u → found := "yes"
- exit inner loop
Since `found = "yes"`, continue with `i := 2`.
---
## 🔁 Second iteration (i = 2)
- `a[2] = v`
**Inner loop: searching b[1..3] for `v`:**
- j = 1 → b[1] = w ≠ v → found = "no"
- j = 2 → b[2] = u ≠ v → found = "no"
- j = 3 → b[3] = v = v → found := "yes"
- exit inner loop
Since `found = "yes"`, continue with `i := 3`.
---
## 🔁 Third iteration (i = 3)
- `a[3] = w`
**Inner loop: searching b[1..3] for `w`:**
- j = 1 → b[1] = w = w → found := "yes"
- exit inner loop
Since `found = "yes"`, continue with `i := 4`.
Now `i = 4 > m = 3`, so the outer loop terminates.
---
## Final Result
`answer = "A ⊆ B"` → ✅ all elements of \(A\) were found in \(B\).
## ✅ **Final Output:**
\[
\boxed{\text{A ⊆ B}}
\]
