Question:
A Boolean function \( F \) of three variables \( X \), \( Y \), and \( Z \) is given as \[ F(X, Y, Z) = (X' + Y + Z) \cdot (X + Y' + Z') \cdot (X' + Y + Z') \cdot (X' Y' Z' + X Y Z') \] Which one of the following is true?
A Boolean function \( F \) of three variables \( X \), \( Y \), and \( Z \) is given as \[ F(X, Y, Z) = (X' + Y + Z) \cdot (X + Y' + Z') \cdot (X' + Y + Z') \cdot (X' Y' Z' + X Y Z') \] Which one of the following is true?
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To simplify Boolean expressions, use Boolean algebra rules like absorption, distribution, and De Morgan's laws.
Updated On: Jan 8, 2026
- \( F(X, Y, Z) = (X + Y + Z') \cdot (X' + Y' + Z') \)
- \( F(X, Y, Z) = (X' + Y) \cdot (X + Y' + Z') \)
- \( F(X, Y, Z) = X' Z' + Y Z' \)
- \( F(X, Y, Z) = X' Y' Z + X Y Z \)
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The Correct Option is C
Solution and Explanation
We are given a Boolean function \( F(X, Y, Z) \) and need to simplify the expression to match one of the options.
Step 1: Analyze the given function.
The given Boolean function is:
\[
F(X, Y, Z) = (X' + Y + Z) \cdot (X + Y' + Z') \cdot (X' + Y + Z') \cdot (X' Y' Z' + X Y Z').
\]
Step 2: Simplify the Boolean expression.
Using Boolean algebra rules (such as absorption, distribution, and De Morgan's laws), simplify the given expression:
\[
F(X, Y, Z) = X' Z' + Y Z'.
\]
Step 3: Conclusion.
The simplified Boolean function is \( F(X, Y, Z) = X' Z' + Y Z' \), which corresponds to option (C).
Final Answer: \( F(X, Y, Z) = X' Z' + Y Z' \).
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