Is the following statement always true Justify your answer I

Is the following statement always true? Justify your answer. If ab\' + [b +b\'(a +bc)]\' = [a + a\'(ac + b)] (a +b\'), then a = b\'.

Solution

LHS: ab\' + [b +b\'(a +bc)]\'

= ab\' + b\' . (b\'(a+bc))\'

= ab\' + b\' . (b + (a+bc)\')

= ab\' + b\' . (b + a\' . (bc)\')

= ab\' + b\' . (b + a\' . (b\' + c\'))

= ab\' + b\' . (b + a\'b\' + a\'c\')

= ab\' + bb\' + a\'b\' + a\'b\'c\'

= ab\' + 0 + a\'b\' + a\'b\'c\'

= b\'(a + a\' + a\'c\')

= b\'(1)

= b\'

RHS: [a + a\'(ac + b)] (a +b\')

= (a + a\'ac + a\'b).(a + b\')

= (a + a\'b).(a + b\')

= aa + ab\' + aa\'b + a\'bb\'

= a + ab\' + 0 + 0

= a + ab\'

= a(1 + b\')

= a(1)

= a

Therefore, a = b\'

Therefore, the statement is always true.