If A, B, C are three sets, then what is A -(B-C) equal to?

Options

A. `A-(B nn C)`
B. `(A-B) uu C`
C. `(A-B) uu (A nn C)`
D. `(A-B) uu (A-C)`

Answer

To solve the problem \( A - (B - C) \), we can break it down step by step.

### Step 1: Understand the Expression
The expression \( A - (B - C) \) involves two operations: set subtraction and the operation inside the parentheses. We need to first evaluate \( B - C \).

**Hint:** Remember that \( B - C \) means the elements that are in set B but not in set C.

### Step 2: Evaluate \( B - C \)
The operation \( B - C \) represents all the elements that are in set B but not in set C. This can be visualized as removing the elements of C from B.

**Hint:** Visualize or sketch the sets to see which elements remain in B after removing those that are also in C.

### Step 3: Substitute Back into the Expression
Now, we substitute \( B - C \) back into the original expression. We need to find \( A - (B - C) \).

**Hint:** This means we are looking for all the elements in A that are not in the result of \( B - C \).

### Step 4: Evaluate \( A - (B - C) \)
To find \( A - (B - C) \), we need to identify the elements in A that are not in the set we just found (which is \( B - C \)).

**Hint:** Again, visualize or sketch the sets to see which elements of A remain after removing those that are in \( B - C \).

### Step 5: Final Expression
The result of \( A - (B - C) \) can be expressed in terms of set operations. We can rewrite this as:
\[ A - (B - C) = A \cap (C \cup (A - B)) \]
This means we are taking the intersection of A with the union of C and the elements that are in A but not in B.

**Hint:** Remember that the intersection operation will give you the elements that are common between the sets.

### Conclusion
Thus, the final result of \( A - (B - C) \) can be expressed as:
\[ A - (B - C) = A \cap (C \cup (A - B)) \]

This means that the elements remaining in A after removing those that are in \( B - C \) can be represented as the intersection of A with the union of C and the elements of A that are not in B.