1. **Define the Sets:**
- Let the set \( A \) represent the address set \( A' \) after performing Fisher-Yates shuffling, containing \( n \) elements.
- Let the set \( C \) represent a randomly chosen set of one-time addresses, containing \( m \) elements.
2. **Intersection Concept:**
- The intersection \( A \cap C \) represents the common elements between sets \( A \) and \( C \). If there is never any overlap between \( A \) and \( C \), then \( A \cap C = \emptyset \).
3. **Theoretical Independence:**
- After performing the Fisher-Yates shuffle, the relationship between address sets \( A \) and \( C \) is randomized. If the distribution of elements is uniform (i.e., any element has an equal probability of appearing in set \( A \)), then theoretically, sets \( A \) and \( C \) should have no inherent relationship.
4. **Probability Analysis:**
- Let \( P(A_i \in C) \) represent the probability that the \( i \)-th address in \( A \) appears in \( C \). If the address sets are uniformly distributed, and there is no correlation between \( A \) and \( C \), then \( P(A_i \in C) = \frac{m}{N} \), where \( N \) is the total number of possible addresses.
- After homogenization, the relationship between \( A \) and \( C \) is eliminated, and the probability \( P(A_i \in C) \) returns to its original uniform distribution.
5. **Mathematical Reasoning:**
- Before shuffling or randomization, sets \( A \) and \( C \) might have had a one-to-one correspondence, meaning their intersection was non-empty. After processing, this relationship is destroyed, and the intersection \( A \cap C \) now depends only on random selection, not on the original correspondence.
- As a result, the probability of intersection is purely based on the distribution of random variables, not on any pre-existing relationship between addresses. Therefore, the intersection \( A \cap C \) is possible, but it no longer holds the original one-to-one correspondence.
1. **定義集合：**
- 設集合 \( A \) 表示經過 Fisher-Yates 洗牌后的地址集合 \( A' \)，其中包含 \( n \) 個元素。
- 設集合 \( C \) 表示隨機選擇的一次性地址集合，包含 \( m \) 個元素。
2. **交集的概念：**
- 交集 \( A \cap C \) 表示在集合 \( A \) 和集合 \( C \) 中共有的元素。如果 \( A \) 和 \( C \) 之間從來沒有交集，則 \( A \cap C = \emptyset \)。
3. **理論上的獨立性：**
- 在進行 Fisher-Yates 洗牌後，地址集合 \( A \) 和 \( C \) 的關係被打亂。如果我們認為每個元素的分佈是均勻的（即任何元素出現在集合 \( A \) 中的概率都是相同的），則 \( A \) 和 \( C \) 在理論上應該是沒有關係的。
4. **概率分析：**
- 令 \( P(A_i \in C) \) 表示 \( A \) 中第 \( i \) 個地址出現在 \( C \) 中的概率。如果地址集合是均勻分佈的，且 \( A \) 和 \( C \) 之間沒有任何關聯，那麼 \( P(A_i \in C) = \frac{m}{N} \)，其中 \( N \) 是所有可能地址的總數。
- 當 \( A \) 和 \( C \) 進行了同質化處理後，它們之間的關聯性消失，且 \( P(A_i \in C) \) 將回到原始的均勻分佈狀態。
5. **數學推論：**
- 在洗牌或隨機化處理之前，集合 \( A \) 和 \( C \) 可能具有一對一的對應關係，這意味著交集是非空的。處理之後，這種關係被消除了，交集 \( A \cap C \) 僅依賴於隨機選擇，沒有固定的對應關係。
- 結果，交集的概率僅僅取決於隨機變量的分佈，而不是地址的原始關聯。因此，交集 \( A \cap C \) 是可能的，但不再具有原本的一對一對應意義。