Simplify the following terms using consensus theorem WYZWXYW

Simplify the following terms using consensus theorem: W\'Y\'Z+W\'XY+WXZ+W\'XZ+XY\'Z+XYZ (a\'+b\'+c)(b+c\'+d\')(a\'+c+d)(a+b\'+d\')(b+c+d)(a+c\'+d\')

Solution

1) w\'y\'z+w\'xy+wxz+w\'xz+xy\'+xyz

=w\'y\'z+w\'xy+xz(w+w\')+xz(y\'+y)    

=w\'y\'z+w\'xy+xz+xz                             (by boolean algebra w+w\'=1)

=w\'y\'z+w\'xy+xz                                   (remove the duplicates by consensus theorem)

=w\'(y\'z+xy)+xz

=w\'(zx)+xz                                          (remove the repeated elements)

=w\'

2) (a\'+b\'+c)(b+c\'+d\')(a\'+c+d\')(a\'+b\'+d)(b+c+d)(a+c\'+d\')

=(a\'+b\'+c)(b+c\'+d\')(a\'+b\'+d\')(b+c+d)+(a+c\'+d\')            (by removing consensus theorem)

=(b\'+d\')+(b\'+d\')+(b+c\')+(b+c)+(a+d)+(a+c\')                 (by using consensus theorem)

=b\'+d\'+b+a

=a+d\'+(b\'+b)                                                            (by boolean expression b\'+b=1)

=a+d\'