Simplify the following boolean functions Show all steps and

Simplify the following boolean functions. Show all steps and list the boolean identity used for each step of simplification. F(x, y) = x\'y\' + xy\' + y(xy\') F(x, y) = x\'y\' + x\'y + xy F(x, y) = x\'y\' + x\'y + xy\' F(x, y, z) = y(x\'z + xz) + x\'yz\' + xy\'z

Solution

1)F(x,y)=x\'y\'+xy\'+y(xy\')

=y\'(x\'+x)+y(xy\')............ complement of x

=y\'(1)+x(yy\')............... assiciative

=y\'+x(0)

F(x,y) =y\'

2)F(x,y)=x\'y\'+x\'y+xy

=x\'(y\'+y)+xy ...... complement of y

=x\'(1)+xy

=x\'+xy

=(x\'+x)(x\'+y) ...... complement of x

=(1)(x\'+y)

F(X,Y)=x\'+y

3)F(x,y)=x\'y\'+x\'y+xy\'

   =x\'(y\'+y)+xy\'

   =x\'(1)+xy\'

   =x\'+xy\'

   =(x\'+x)(x\'+y\')

   =(1)(x\'+y\')

   =x\'+y\'

4)F(x,y,z)=y(x\'z+xz)+x\'yz\'+xy\'z

   =y\'+(x\'z+xz)\'+x\'yz\'+xy\'z

   =y\'+(x\'\'+z\')(x\'+z\')+x\'yz\'+xy\'z

   =y\'+(x+z\')(x\'+z\')+(x\'yz\'+xy\'z)\'

   =y\'+(x+z\')(x\'+z\')+(x\'yz\')\'(xy\'z)\'

   =y\'+(x+z\')(x\'+z\')+(x\'\'+y\'+z\'\')(x\'+y\'\'+z\')

   =y\'+(x+z\')(x\'+z\')+(x+y\'+z)(x\'+y+z\')