If A and B are subsets of a set X, then what is the value of `{A cap (X-B)} cup B` ?

Options

A. `A cup B`
B. `A cap B`
C. A
D. B

Answer

To solve the problem, we need to simplify the expression `{A ∩ (X - B)} ∪ B`. Let's break it down step by step.

### Step 1: Understand the components of the expression
- **A** and **B** are subsets of a set **X**.
- **X - B** (or **X \ B**) represents the complement of **B** in **X**, which includes all elements in **X** that are not in **B**.

**Hint for Step 1:** Remember that the complement of a set includes all elements not in that set.

### Step 2: Find the intersection A ∩ (X - B)
- The expression **A ∩ (X - B)** means we are looking for elements that are in **A** and also in **X** but not in **B**. This can also be expressed as **A ∩ B'**, where **B'** is the complement of **B**.

**Hint for Step 2:** The intersection will give you the elements that are in both sets.

### Step 3: Combine with B using the union
- Now we need to take the union of the result from Step 2 with **B**. So we have:
\[ (A ∩ (X - B)) ∪ B \]
This means we are combining all elements that are either in **A** but not in **B**, or in **B**.

**Hint for Step 3:** The union combines all unique elements from both sets.

### Step 4: Analyze the result
- The union of **(A ∩ (X - B))** and **B** can be interpreted as taking all elements from **A** that are not in **B** and adding all elements from **B**. This means that any element in **B** will be included in the final result regardless of whether it is in **A** or not.

### Step 5: Conclusion
- Therefore, the final result simplifies to:
\[ A ∪ B \]
This means that the value of the expression `{A ∩ (X - B)} ∪ B` is simply the union of sets **A** and **B**.

**Final Answer:**
\[ A ∪ B \]