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Answer

To solve the equation \( A - (B \cup C \cup D) = (A - B) \cap (A - C) \cap (A - D) \), we will break it down step by step.

### Step 1: Understand the left-hand side
The left-hand side of the equation is \( A - (B \cup C \cup D) \). This expression represents the set of elements that are in set \( A \) but not in the union of sets \( B \), \( C \), and \( D \).

### Step 2: Rewrite the left-hand side using set notation
Using set notation, we can express this as:
\[ A - (B \cup C \cup D) = \{ x \in A \mid x \notin (B \cup C \cup D) \} \]
This means we are looking for all elements \( x \) that are in \( A \) and not in either \( B \), \( C \), or \( D \).

### Step 3: Understand the right-hand side
The right-hand side of the equation is \( (A - B) \cap (A - C) \cap (A - D) \). This expression represents the intersection of three sets:
- \( A - B \): elements in \( A \) but not in \( B \)
- \( A - C \): elements in \( A \) but not in \( C \)
- \( A - D \): elements in \( A \) but not in \( D \)

### Step 4: Rewrite the right-hand side using set notation
Using set notation, we can express this as:
\[ (A - B) \cap (A - C) \cap (A - D) = \{ x \in A \mid x \notin B \} \cap \{ x \in A \mid x \notin C \} \cap \{ x \in A \mid x \notin D \} \]
This means we are looking for all elements \( x \) that are in \( A \) and not in \( B \), not in \( C \), and not in \( D \).

### Step 5: Compare both sides
Now we can see that both sides of the equation represent the same set:
- The left-hand side \( A - (B \cup C \cup D) \) is the set of elements in \( A \) that are not in \( B \), \( C \), or \( D \).
- The right-hand side \( (A - B) \cap (A - C) \cap (A - D) \) is also the set of elements in \( A \) that are not in \( B \), not in \( C \), and not in \( D \).

### Conclusion
Since both sides represent the same set, we can conclude that:
\[ A - (B \cup C \cup D) = (A - B) \cap (A - C) \cap (A - D) \]
Thus, the equation is verified.