Which of the following operations is functionally complete ?
Consider the operations
$\textit{f (X, Y, Z) = X'YZ + XY' + Y'Z'}$ and $\textit{g (X, Y, Z) = X'YZ + X'YZ' + XY}$
Is there Any technique to approach such questions ?
Which of the following operations is functionally complete?
0
$\begingroup$
functions
discrete-mathematics
1 Answers
1
Hint: Try to convert the function into a nand or a nor function. If it gets converted then it is functionally complete.
The first function we have is $$f(X,Y,Z)=X'YZ+XY'+Y'Z'\tag{1.}$$
How equation 1 can be converted into $Y'Z'$
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Put $X=0$ $\to f=YZ+Y'Z'$
multiply by Y' then $f.Y'=Y'Z' \tag{2.}$
from equation 2 ,it is clear that if we can get complement of $Y'$ and $AND$ function from function $f$ then we can get Nor function also.
Put $Y=Z=1 \to f=X'$ put $X=Y$ so $f=Y'$. Put $Y=1, X=X'$
in $f\to f=XZ$ that's how we get both AND and NOT function to eventually get every other function.
Similarly try with other function.