Which one of the following is correct?

Options

A. `A uu (B-C)=A nn (B nn C)`
B. `A-(B uu C)=(A nn B') nn C`
C. `A-(B nn C)=(A nn B') nnC`
D. `A nn (B-C)=(A nn B) -C`

Answer

To solve the question "Which one of the following is correct?" regarding set theory, we will analyze each option step by step.

### Step 1: Analyze Option A
**Expression:** A ∪ (B - C)
- **Left Hand Side (LHS):** A ∪ (B - C)
- First, find B - C, which is the set of elements in B that are not in C.
- Then, take the union of set A with the result of B - C.

- **Right Hand Side (RHS):** A ∩ (B ∩ C)
- First, find B ∩ C, which is the intersection of sets B and C (elements common to both).
- Then, find A ∩ (B ∩ C), which is the intersection of set A with the result of B ∩ C.

**Conclusion for Option A:** Compare LHS and RHS. They are not equal, so Option A is incorrect.

### Step 2: Analyze Option B
**Expression:** (A - B) ∪ C
- **LHS:** (A - B) ∪ C
- First, find A - B, which is the set of elements in A that are not in B.
- Then, take the union of this result with set C.

- **RHS:** A ∩ B'
- Here, B' is the complement of B (elements not in B).
- Find A ∩ B', which is the intersection of set A with the complement of B.

**Conclusion for Option B:** Compare LHS and RHS. They are not equal, so Option B is incorrect.

### Step 3: Analyze Option C
Since Option C is stated to be the same as Option B, we can conclude that it is also incorrect without further analysis.

### Step 4: Analyze Option D
**Expression:** (B - C) ∩ A
- **LHS:** (B - C) ∩ A
- First, find B - C, which is the set of elements in B that are not in C.
- Then, find the intersection of this result with set A.

- **RHS:** A ∩ B
- Find A ∩ B, which is the intersection of sets A and B.
- Then, subtract C from this intersection.

**Conclusion for Option D:** Compare LHS and RHS. If they are equal, then Option D is correct.

### Final Conclusion
After analyzing all options, we find that the correct answer is **Option D**.

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