What will be the input of `A` and `B` for the Boolean expression `bar((A+B)).bar((A.B))=1`?

Options

A. (0,0)
B. (0,1)
C. (1,0)
D. (1,1)

Explanation

To solve the Boolean expression \(\overline{(A + B)} \cdot \overline{(A \cdot B)} = 1\), we will follow these steps:

### Step 1: Rewrite the Expression
We start with the given expression:
\[ \overline{(A + B)} \cdot \overline{(A \cdot B)} = 1 \]
This means that both \(\overline{(A + B)}\) and \(\overline{(A \cdot B)}\) must be true (equal to 1) for the entire expression to equal 1.

### Step 2: Analyze \(\overline{(A + B)}\)
The expression \(\overline{(A + B)}\) is true when both \(A\) and \(B\) are false (0). This is because:
- If \(A = 0\) and \(B = 0\), then \(A + B = 0\) and \(\overline{(A + B)} = 1\).
- If either \(A\) or \(B\) is 1, then \(A + B = 1\) and \(\overline{(A + B)} = 0\).

### Step 3: Analyze \(\overline{(A \cdot B)}\)
The expression \(\overline{(A \cdot B)}\) is true when at least one of \(A\) or \(B\) is false (0). This is because:
- If \(A = 0\) or \(B = 0\), then \(A \cdot B = 0\) and \(\overline{(A \cdot B)} = 1\).
- If both \(A\) and \(B\) are 1, then \(A \cdot B = 1\) and \(\overline{(A \cdot B)} = 0\).

### Step 4: Combine the Conditions
For the entire expression to be true (\(= 1\)), both conditions must hold:
1. \(\overline{(A + B)} = 1\) implies \(A = 0\) and \(B = 0\).
2. \(\overline{(A \cdot B)} = 1\) is satisfied if at least one of \(A\) or \(B\) is 0.

Since the first condition requires both \(A\) and \(B\) to be 0, this will also satisfy the second condition.

### Conclusion
Thus, the only inputs for \(A\) and \(B\) that satisfy the Boolean expression \(\overline{(A + B)} \cdot \overline{(A \cdot B)} = 1\) are:
\[ A = 0, B = 0 \]