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The problem in the image asks about how many pairs \((A, B)\) satisfy a specific condition, given a set \(X = \{3, -4, 5\}\) and two subsets \(A\) and \(B\), with the condition that the number of elements in \(B\) is less than the number of elements in \(A\), and the union \(\{3, -4\} \cup (A \setminus B) = X\).

### Step-by-step approach:

1. **Set \(X = \{3, -4, 5\}\)**:
   - The elements of set \(X\) are \(3, -4, 5\).

2. **Condition: \(\{3, -4\} \cup (A \setminus B) = X\)**:
   - This means that combining \(\{3, -4\}\) with the set difference \(A \setminus B\) should result in \(X = \{3, -4, 5\}\). Since \(X\) contains all the elements, \(A \setminus B\) must include the element \(5\) to make up for the missing element in \(\{3, -4\}\).

3. **Subsets \(A\) and \(B\)**:
   - Subset \(A\) must contain enough elements such that when you remove the elements of \(B\), the result satisfies the condition of completing \(X\).
   - Subset \(B\) contains fewer elements than \(A\).

4. **Constructing the pairs \((A, B)\)**:
   - Set \(A\) must contain the element \(5\) because we need \(A \setminus B\) to provide \(5\).
   - Subset \(A\) can also include either or both of the elements \(3\) and \(-4\), but this is optional because they are already present in the set \(\{3, -4\}\).
   - Subset \(B\) must include elements from \(A\), but it cannot contain as many elements as \(A\) (since \(|B| < |A|\)).

5. **Counting the possibilities**:
   - The possible subsets for \(A\) are:
     - \(A = \{5\}\)
     - \(A = \{3, 5\}\)
     - \(A = \{-4, 5\}\)
     - \(A = \{3, -4, 5\}\)
   - For each subset \(A\), we need to select \(B\) such that \(|B| < |A|\) and \(B \subseteq A\).

6. **Finding the pairs**:
   - If \(A = \{5\}\), there are no valid subsets \(B\) because \(B\) must have fewer elements, but the empty set does not work under the given conditions.
   - If \(A = \{3, 5\}\), \(B\) can be \(\emptyset\) or \(\{3\}\).
   - If \(A = \{-4, 5\}\), \(B\) can be \(\emptyset\) or \(\{-4\}\).
   - If \(A = \{3, -4, 5\}\), \(B\) can be \(\emptyset\), \(\{3\}\), \(\{-4\}\), or \(\{3, -4\}\).

### Conclusion:
Now, summing up the possibilities for valid pairs \((A, B)\):
- For \(A = \{5\}\), no valid \(B\).
- For \(A = \{3, 5\}\), there are 2 possibilities for \(B\).
- For \(A = \{-4, 5\}\), there are 2 possibilities for \(B\).
- For \(A = \{3, -4, 5\}\), there are 4 possibilities for \(B\).

Thus, the total number of valid pairs \((A, B)\) is \(0 + 2 + 2 + 4 = 8\).

### Final answer:
There are **8 pairs \((A, B)\)** that satisfy the given condition.

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Would you like more details or have any other questions?

Here are 5 related questions for further exploration:
1. How does the set difference \(A \setminus B\) operate in other set theory problems?
2. How does the condition \(|B| < |A|\) influence the number of possible pairs?
3. Can you find other examples where unions of set differences are used in real applications?
4. What are the basic properties of set operations like union, intersection, and difference?
5. How would the solution change if the set \(X\) contained more or fewer elements?

**Tip:** When dealing with set operations, always break down the conditions step by step, visualizing how subsets combine to form the target set.

Cho tập hợp X = {3, -4, 5} có hai tập con A và B (số phần tử của tập B ít hơn số phần tử của tập A). Có bao nhiêu cặp (A, B) mà {3, -4} ∪ (A \ B) = X?

Solve a set theory problem involving set X = {3, -4, 5}, with conditions on subsets A and B. Learn how to calculate the number of pairs (A, B) where the union of set differences results in the original set X. Suitable for high school students learning set operations.

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