Prove that A−(B∪C)=(A−B)∩(A−C),for all sets A,B,C.

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.**

---

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?



Learn how to prove the set identity A - (B ∪ C) = (A - B) ∩ (A - C) using set theory. This detailed proof covers key set operations and logical reasoning. Suitable for students in Grades 10-12 studying set theory.

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