If A and B are two sets, then `(A uu B') nn (A' nn B)` is equal to

Options

A. A'
B. A
C. B'
D. None of these

Explanation

To solve the problem `(A ∪ B') ∩ (A' ∩ B)`, we will use set theory operations step by step.

### Step 1: Understand the components of the expression
- **A**: A set
- **B**: Another set
- **B' (B complement)**: The elements not in set B
- **A' (A complement)**: The elements not in set A
- **A ∪ B'**: The union of set A and the complement of set B
- **A' ∩ B**: The intersection of the complement of set A and set B

### Step 2: Find A ∪ B'
- The union of A and B' includes all elements that are in A or not in B. This means we take all elements from A and all elements outside of B.

### Step 3: Find A' ∩ B
- The intersection of A' and B includes all elements that are in B but not in A. This means we take only those elements that are in B and exclude any elements that are also in A.

### Step 4: Find the intersection of the two results
- Now we need to find `(A ∪ B') ∩ (A' ∩ B)`. This means we are looking for elements that are common to both sets we found in Steps 2 and 3.

### Step 5: Analyze the intersection
- The first set, `A ∪ B'`, includes elements in A and elements outside of B.
- The second set, `A' ∩ B`, includes elements that are in B but not in A.
- The intersection of these two sets will be empty because the elements in `A' ∩ B` cannot be in `A` (by definition of A') and thus cannot be in `A ∪ B'` (since `A ∪ B'` includes elements from A).

### Conclusion
Therefore, the result of `(A ∪ B') ∩ (A' ∩ B)` is the empty set, denoted as `∅`.

### Final Answer
**(A ∪ B') ∩ (A' ∩ B) = ∅**

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