write an expression for the complement of F . 4) F(w, x, y, z) = xyz’(y’z + x)’ + (w’yz + x’)
The complement is the inverse of a variable and is indicated by a bar over variable or dash sign such as x'.
De Morgan’s law states that (x + y)' = x' . y' and ( x . y)' = x' + y'
We can find the complement of the function using the above two rule stated by De Morgan’s Law.
Given:
F = xyz’(y’z + x)’ + (w’yz + x’)
Complement of F i.e. F' = (xyz’(y’z + x)’ + (w’yz + x’))'
= (xyz’(y’z + x)’)' . ((w’yz + x’))'
= [ (xyz')' + (y'z + x) ] . [ (w'yz)' . x] [By using involution law: (x')'=x]
= [(x' + y' + z) + (y'z + x)].[(w+y'+z').x]
= (x' + y' + z + y'z + x)(wx + xy' + xz')
= (x + x' + y' + z + y'z) (wx + xy' + xz') [Using Complement law: x + x'=1]
= (1 + y' + z + y'z) (wx + xy' + xz')
= (1+y') + z (1 + y') (wx + xy' + xz') [Using Distributive law: x(y+z) = xy+xz]
= (1 + y') (1 + z) (wx + xy' + xz')
= 1.1 (wx + xy' + xz') [Using Identity Law: 1+x=1]
= (wx + xy' + xz')
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