Let U be the universal set B, C are sub-sets of U. If A = `B cup C`, then U -(U - ((U-A')))) is equal to

Options

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

Answer

To solve the problem step by step, we need to analyze the expression given: \( U - (U - (U - A')) \), where \( A = B \cup C \).

### Step 1: Understand the components
- Let \( A = B \cup C \).
- The complement of \( A \) with respect to the universal set \( U \) is denoted as \( A' \) or \( U - A \).

### Step 2: Substitute \( A \) into the expression
- We can rewrite the expression:
\[ U - (U - (U - A')) \]
becomes:
\[ U - (U - (U - (U - (B \cup C)))) \]

### Step 3: Simplify \( U - A' \)
- Since \( A' = U - A = U - (B \cup C) \), we can substitute this into the expression:
\[ U - (U - (U - (U - (B \cup C)))) \]

### Step 4: Simplify \( U - A' \)
- Now we simplify \( U - (U - (B \cup C)) \):
- The expression \( U - (U - (B \cup C)) \) gives us \( B \cup C \) because it represents the elements in \( U \) that are not outside \( B \cup C \).

### Step 5: Substitute back into the expression
- Now we have:
\[ U - (U - (B \cup C)) \]
which simplifies to \( B \cup C \).

### Step 6: Final simplification
- Finally, we have:
\[ U - (B \cup C) \]
which represents the elements in \( U \) that are not in \( B \cup C \).

### Conclusion
Thus, the final result of the expression \( U - (U - (U - A')) \) is:
\[ B \cup C \]