For any two sets A and B, the value of `[(A-B) cup B]` is equal to

Options

A. `A cap B`
B. `A cap B`
C. A-B
D. B-A

Answer

To find the value of \((A - B) \cup B\), we can follow these steps:

### Step 1: Understand the expression \(A - B\)
The expression \(A - B\) represents the set of elements that are in set \(A\) but not in set \(B\). This can also be expressed using set notation as:
\[ A - B = A \cap B' \]
where \(B'\) is the complement of \(B\).

### Step 2: Substitute \(A - B\) in the expression
Now, we can substitute \(A - B\) in our original expression:
\[ (A - B) \cup B = (A \cap B') \cup B \]

### Step 3: Apply the distributive property
Next, we can apply the distributive property of sets:
\[ (A \cap B') \cup B = A \cup B \]
This means that we are taking the union of the elements in \(A\) (not in \(B\)) and all elements in \(B\).

### Step 4: Simplify the expression
Now, we need to simplify the expression. We know that:
\[ B' \cup B = U \]
where \(U\) is the universal set. Therefore, the union of \(B\) and its complement covers all elements in the universal set.

### Step 5: Conclusion
Thus, we can conclude that:
\[ (A - B) \cup B = A \cup B \]

### Final Answer
The value of \((A - B) \cup B\) is equal to \(A \cup B\).

---