In the Boolean algebra `bar(A).bar(B)` equals is :-

Options

A. A + B
B. `bar(A+B)`
C. A . B
D. `bar(A).B`

Answer

To solve the Boolean algebra expression \( \overline{A} \cdot \overline{B} \) and find its equivalent expression, we can use De Morgan's theorem. Here’s a step-by-step breakdown of the solution:

### Step 1: Understand De Morgan's Theorem
De Morgan's theorem states that:
1. The complement of a product is equal to the sum of the complements:
\[ \overline{A \cdot B} = \overline{A} + \overline{B} \]
2. The complement of a sum is equal to the product of the complements:
\[ \overline{A + B} = \overline{A} \cdot \overline{B} \]

### Step 2: Apply De Morgan's Theorem
We are given the expression \( \overline{A} \cdot \overline{B} \). According to De Morgan's theorem, we can rewrite this as:
\[ \overline{A} \cdot \overline{B} = \overline{A + B} \]
This means that the expression \( \overline{A} \cdot \overline{B} \) is equivalent to the complement of the sum of \( A \) and \( B \).

### Step 3: Identify the Equivalent Expression
Now, we need to find the equivalent expression for \( \overline{A} \cdot \overline{B} \):
- From our application of De Morgan's theorem, we have:
\[ \overline{A} \cdot \overline{B} = \overline{A + B} \]

### Step 4: Check the Options
Now, we will check the options provided:
1. \( A + B \) (not correct)
2. \( \overline{A + B} \) (correct)
3. \( A + B \) (not correct)
4. Other options (not provided)

From the above analysis, the equivalent expression for \( \overline{A} \cdot \overline{B} \) is:
\[ \overline{A + B} \]

### Final Answer
Thus, \( \overline{A} \cdot \overline{B} \) equals \( \overline{A + B} \).

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