Which of the following is/are correct relations in Boolean algebra?

Options

A. `A.B+barA.barB=((A.B)barA.barB)`
B. `A.B+barA.barB=(bar(A.B))(barA.barB)`
C. `bar(A.B+barA.barB)=(bar(A.B)).(bar(barA)+bar(barB))`
D. `bar(A.B+barA.B)=(A.B).(bar(barA)+bar(barB))`

Explanation

To determine which of the given relations in Boolean algebra is correct, we can analyze the options based on De Morgan's theorem and other Boolean algebra properties. Let's go through the steps systematically.

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

### Step 2: Analyze Each Option
Assuming we have options A, B, C, and D, we need to check each option against De Morgan's theorem and other Boolean identities.

#### Option A:
Check if it follows the rules of De Morgan's theorem or any other Boolean algebra property.

#### Option B:
Similarly, verify this option against the established rules.

#### Option C:
This option should be checked thoroughly, as the video suggests it might be the correct one.

#### Option D:
Again, verify this option against the rules.

### Step 3: Identify the Correct Relation
After analyzing each option, we find that:
- **Option C** aligns with the derived relations from De Morgan's theorem and does not contradict any Boolean algebra rules.

### Conclusion
The correct relation in Boolean algebra, based on the analysis, is **Option C**.

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