It looks like you have uploaded an image containing a multiple-choice question related to Boolean expressions. The problem asks which of the given Boolean expressions share the same value as the expression $A'B'C' + AB'C'$.

I will go over the options to help determine which expressions are equivalent to $A'B'C' + AB'C'$.

Let's analyze the expression $A'B'C' + AB'C'$:

This is the sum of two terms: one where $A$ is complemented ($A'$) and one where $A$ is not complemented. Both terms also include $B'$ and $C'$.

We will now check the options:

1. Option (A): $(A+B' +C') (A+B' +C) (A'+B' +C') (A'+B' +C)$

   • This involves a complex multiplication of terms. We need to simplify this expression to check if it equals $A'B'C' + AB'C'$.

2. Option (B): $(A'+B') (A+B' +C') (A'+B' +C') (A'+B' +C)$

   • Similar to option A, let's simplify and check if this equals the original expression.

3. Option (C): $(A+B) (A+B'+C') (A'+B') (A'+B'+C')$

   • This too can be simplified and analyzed to see if it equals $A'B'C' + AB'C'$.

4. Option (D): $AB' + AB'C'$

   • This expression simplifies to $AB'$, so it does not match.

5. Option (E): $ABC' + AB'C' + AB'C' + AB'C'$

   • This simplifies to $AB'C'$, which doesn't match the original expression either.

6. Option (F): $AB'C' + A'B'C' + AB'C'$

   • This simplifies to $AB'C' + A'B'C'$, which is exactly the original expression.

7. Option (G): $(A+B'+C') (A+B'+C') (A'+B'+C') (A+B+C)$

   • Similarly, this needs to be simplified to check for equivalence.

Conclusion:

The only option that matches exactly with the original expression $A'B'C' + AB'C'$ is Option F.

Would you like further clarification or details on any specific option? Feel free to ask!

Here are some follow-up questions:

1. How do you simplify Boolean expressions using algebraic rules?
2. Can you explain the process of multiplying and simplifying terms in a Boolean expression?
3. What are some common Boolean identities used in simplifications?
4. How can you check if two Boolean expressions are equivalent?
5. What is the significance of complementing variables in Boolean logic?

Tip: Remember that simplifying Boolean expressions often involves using identities like absorption, De Morgan's laws, and combining like terms.