Define the connective * for the Boolean variables X and Y as: X * Y = XY + X'…
2007
Define the connective * for the Boolean variables X and Y as: X * Y = XY + X' Y'. Let Z = X * Y.
Consider the following expressions P, Q and R.
P: X = Y⋆Z
Q: Y = X⋆Z
R: X⋆Y⋆Z=1Which of the following is TRUE?
- A.
Only P and Q are valid
- B.
Only Q and R are valid.
- C.
Only P and R are valid.
- D.
All P, Q, R are valid.
Attempted by 470 students.
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Correct answer: D
Key idea: The connective * is XNOR: X * Y = 1 when X = Y, and 0 when X ≠ Y. Given Z = X * Y.
P and Q:
For X = Y * Z: If Z = 1 then X = Y, so Y * Z = Y * 1 = Y = X. If Z = 0 then X ≠ Y, so Y * Z = Y XNOR 0 = NOT(Y) = X. Thus X = Y * Z holds in both cases.
For Y = X * Z: The argument is symmetric to the previous one, so Y = X * Z also holds for all X,Y.
R: (X * Y) * Z = Z * Z = 1 because Z = X * Y, and any value XNOR itself equals 1.
Conclusion: All three expressions are true for every assignment of X and Y, so all P, Q and R are valid.
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