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 solve the problem of finding the value of \((A - B) \cup B\), we can follow these steps:

### Step 1: Understand the operation \(A - B\)

The operation \(A - B\) represents the set of elements that are in set \(A\) but not in set \(B\). This can be visualized using a Venn diagram where we shade the part of set \(A\) that does not overlap with set \(B\).

**Hint:** Remember that \(A - B\) is all the elements in \(A\) excluding those that are also in \(B\).

### Step 2: Visualize \(A - B\)

In a Venn diagram, if we have two circles representing sets \(A\) and \(B\), the area of set \(A\) that does not overlap with set \(B\) is what we denote as \(A - B\).

**Hint:** Drawing a Venn diagram can help you visualize the elements of \(A\) and \(B\) clearly.

### Step 3: Apply the union operation

Now we need to find \((A - B) \cup B\). The union operation \(\cup\) combines all elements from both sets. This means we take all elements from \(A - B\) and add all elements from \(B\).

**Hint:** The union of two sets includes every element that is in either set.

### Step 4: Combine the sets

Since \(A - B\) includes all elements in \(A\) that are not in \(B\), and \(B\) includes all elements in \(B\), when we take the union, we will have all elements from \(A\) (excluding those in \(B\)) plus all elements from \(B\).

This can be simplified to just \(A \cup B\) because any element in \(B\) is already included in the union.

**Hint:** Think about how union works: if an element is in either set, it will be included in the result.

### Step 5: Conclusion

Thus, we conclude that \((A - B) \cup B = A \cup B\).

**Final Answer:** \((A - B) \cup B = A \cup B\)