Multiply out the Boolean expression (A' + B)(A' + C)(A' + D).
Understand the Problem
The question is asking for the multiplication of a given Boolean expression, specifically (A' + B)(A' + C)(A' + D). The task is to simplify or expand this expression to find the correct representation among the provided options.
Answer
$A' + BCD$
Answer for screen readers
The final answer is $A' + BCD$.
Steps to Solve
- Distribute the First Two Terms
Multiply the first two terms $(A' + B)(A' + C)$:
$$(A' + B)(A' + C) = A' A' + A' C + B A' + B C = A' + A' C + B A' + B C$$
Since $A' A' = A'$, this simplifies to:
$$A' + A' C + B C$$
This can be further simplified as $A' + B C$ because $A' C$ is absorbed by $A'$.
- Distribute with the Third Term
Now multiply the result with the third term $(A' + D)$:
$$(A' + B C)(A' + D)$$
Distributing gives:
$$(A' + B C) A' + (A' + B C) D = A' + A' D + B C A' + B C D$$
Since $A' + A' D = A'$ (absorption law), we get:
$$A' + B C D$$
So, we can further simplify to:
$$A' + B C D$$
- Final Expression
The final expanded form of the expression is:
$$A' + B C D$$
The final answer is $A' + BCD$.
More Information
This simplified Boolean expression indicates the states where the result is true. $A'$ represents cases where $A$ is false, while $BCD$ details when $B$, $C$, and $D$ are all true.
Tips
- Forgetting to apply absorption laws, which can lead to unnecessary complexity.
- Mismanaging the order of distribution, which can result in incorrect combinations.
- Not recognizing that $A' + A'X = A'$ for any variable $X$.
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