To prove that:

$A - (B \cup C) = (A - B) \cap (A - C)$

for all sets $A, B, C$, we use set notation and logical reasoning.

Step 1: Definition of Set Difference

The set difference $A - X$ is defined as:

$A - X = \{ x \in A \mid x \notin X \}$

Using this, we rewrite both sides of the equation in terms of set membership.

Step 2: Prove $A - (B \cup C) \subseteq (A - B) \cap (A - C)$

Let $x \in A - (B \cup C)$. By definition of set difference:

$x \in A \quad \text{and} \quad x \notin B \cup C.$

Since $x \notin B \cup C$, it follows that:

$x \notin B \quad \text{and} \quad x \notin C.$

Thus, by the definition of set difference:

$x \in A - B \quad \text{and} \quad x \in A - C.$

Since $x$ is in both $A - B$ and $A - C$, it must also be in their intersection:

$x \in (A - B) \cap (A - C).$

This proves that:

$A - (B \cup C) \subseteq (A - B) \cap (A - C).$

Step 3: Prove $(A - B) \cap (A - C) \subseteq A - (B \cup C)$

Let $x \in (A - B) \cap (A - C)$. This means:

$x \in A - B \quad \text{and} \quad x \in A - C.$

By the definition of set difference, we know that:

$x \in A, \quad x \notin B, \quad \text{and} \quad x \notin C.$

Since $x \notin B$ and $x \notin C$, it follows that $x \notin B \cup C$, meaning:

$x \in A - (B \cup C).$

Thus, we have shown that:

$(A - B) \cap (A - C) \subseteq A - (B \cup C).$

Step 4: Conclusion

Since we have proved both subset inclusions:

$A - (B \cup C) \subseteq (A - B) \cap (A - C)$

and

$(A - B) \cap (A - C) \subseteq A - (B \cup C),$

it follows that:

$A - (B \cup C) = (A - B) \cap (A - C).$

Thus, the given set identity is proven.

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Would you like further details or have any questions? Here are some related questions to explore:

1. How does this proof change if we consider the union $(A - B) \cup (A - C)$?
2. Can we generalize this proof for more than two sets, i.e., $A - (B_1 \cup B_2 \cup \dots \cup B_n)$?
3. What is the analogous property for set intersection instead of union?
4. How does this relate to De Morgan's laws in set theory?
5. Can this proof be visualized using a Venn diagram?