If A, B and C are three sets, then A-(B-C) equals to

Options

A. A) `A-(B cap C)`
B. B) `(A-B) cup C`
C. C) `(A-B) cup (A cap C)`
D. D) `(A-B) cup (A-C)`

Answer

To solve the problem of finding \( A - (B - C) \), we will follow these steps:

### Step 1: Understand the operation \( B - C \)
The operation \( B - C \) means we are taking all the elements in set \( B \) and removing any elements that are also in set \( C \). This can be visualized as the part of \( B \) that does not overlap with \( C \).

**Hint:** Think of \( B - C \) as the elements that are in \( B \) but not in \( C \).

### Step 2: Visualize the sets
Draw a Venn diagram with three overlapping circles representing sets \( A \), \( B \), and \( C \). This will help you see the relationships between the sets clearly.

**Hint:** Label the areas where the sets overlap and where they do not to keep track of the elements.

### Step 3: Find \( A - (B - C) \)
Now, we need to find \( A - (B - C) \). This means we take all the elements in set \( A \) and remove the elements that are in \( B - C \).

**Hint:** Remember that \( B - C \) is the part of \( B \) that does not include any elements from \( C \).

### Step 4: Rewrite the expression
We can rewrite the expression \( A - (B - C) \) using set operations. The expression can be rewritten as:
\[ A - (B - C) = A \cap (B - C)' \]
where \( (B - C)' \) is the complement of \( B - C \) with respect to the universal set.

**Hint:** The complement of a set includes everything that is not in that set.

### Step 5: Simplify the expression
Using the properties of set operations, we can express \( A - (B - C) \) as:
\[ A - (B - C) = A \cap (C \cup (A - B)) \]
This means we are taking all elements in \( A \) that are either in \( C \) or not in \( B \).

**Hint:** Think about how the union and intersection work together to combine elements from different sets.

### Step 6: Final expression
Thus, the final expression for \( A - (B - C) \) can be simplified to:
\[ A - (B - C) = (A - B) \cup (A \cap C) \]
This means the result is the union of the elements in \( A \) that are not in \( B \) and the elements that are in both \( A \) and \( C \).

**Hint:** The union combines all unique elements from both sets, ensuring no duplicates.

### Conclusion
The final answer is:
\[ A - (B - C) = (A - B) \cup (A \cap C) \]