According to de morgan's theorem-

Options

A. `overline(A+B+C)=overline(A)+overline(B)+overline(C)`
B. `overline(ABC)=overline(ABC)`
C. `overline(ABC)=overline(A)+overline(B)+overline(C)`
D. `overline(A+B+C)=ABC`

Answer

To solve the question regarding De Morgan's theorem, we will follow these steps:

### Step 1: Understand De Morgan's Theorem
De Morgan's theorem consists of two fundamental laws that describe the relationship between conjunctions (AND operations) and disjunctions (OR operations) in logic. The two laws are:
1. The complement of the conjunction of two variables is equal to the disjunction of their complements.
\[ \overline{A \cdot B} = \overline{A} + \overline{B} \]
2. The complement of the disjunction of two variables is equal to the conjunction of their complements.
\[ \overline{A + B} = \overline{A} \cdot \overline{B} \]

### Step 2: Apply De Morgan's Theorem to Three Variables
Now, let's extend this to three variables, say A, B, and C. According to De Morgan's theorem:
1. The complement of the product of three variables is equal to the sum of the complements of those variables:
\[ \overline{A \cdot B \cdot C} = \overline{A} + \overline{B} + \overline{C} \]
2. The complement of the sum of three variables is equal to the product of their complements:
\[ \overline{A + B + C} = \overline{A} \cdot \overline{B} \cdot \overline{C} \]

### Step 3: Identify the Correct Option
Given the question asks which options are correct according to De Morgan's theorem, we would analyze the options provided (not listed in the question). However, based on the application of De Morgan's theorem, we can conclude that:
- If an option states that the complement of the product of A, B, and C equals the sum of their complements, it is correct.
- If an option states that the complement of the sum of A, B, and C equals the product of their complements, it is also correct.

### Conclusion
After applying De Morgan's theorem to both two and three variables, we can identify the correct options based on the laws stated above.