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I am having a little trouble getting the result on the right-hand side. I keep on evaluating down to $abc$, instead of $bc$. Here is my work below:

$(a'b' + c)(a + b)(b' + a'c')'$

$=(aa'b' + ac + bc + a'bb')(b' + a'c')'$ [as $aa' = 0$]

$=(ac + bc)(b' + a'c')'$

$=(ac + bc)(abc)$ [by DeMorgan's Law s.t. $(a + b)' = a'b'$]

$=aabcc + abbcc$

$=abc$ [by Idempotency s.t. $aa = a$, and $a + a = a$]

Could someone help me to correct my mistake as I should be getting $bc$, instead of $abc$?

Thanks

** Edit: Nevermind. I have found the solution out by using some more principle Boolean algebra identities (as shown below) **

= (ac + bc)(b' + a'c')'

= (ac + bc)(b' + (a + c)')' [using DeMorgan's Law s.t. (a + b)' =a'b']

= (ac + bc)(b' + x')' [using Substitution s.t. x = A + c]

= (ac + bc)((bx)')' [using DeMorgan's Law s.t. a' + b' = (ab)']

= (ac + bc)(bx) [using Involution s.t. (a')' = a]

= (ac + bc)(b(a + c))

= (ac + bc)(ab + bc)

= (aabc + abbc + abcc + bbcc)

= (abc + abc + abc + bc) [using Idempotency s.t. aa = a]

= (abc + bc) [using Idempotency s.t. a + a = a]

= (ay + y) = (y + ya) [using Substitution s.t. y = bc]

= y [using Absorption s.t. a + ab = a]

= bc

  • 0
    When you're using DeMorgan's law, remember that it also implies $(a'c')' = a+c$ – 2017-01-20
  • 0
    As Arthur says: You were careless with the transformation from $(b' + a'c')'$. That should be $(b')'(a'c')'$, which is not $abc$ (expand the second factor correctly, also using DeMorgan's laws, as Arthur has shown). – 2017-01-20

2 Answers 2

1

A little shorter method to help you out: $$ (a'b'+c)(a+b)(b'+a'c')' = b(a+c)(a+b)(a'b'+c) = b(a+c)(a'b'+c) = b(c)(a+c) = bc $$

First equality is by De Morgan on $(a'b'+c)'$, second is by absorption ($b(a+b) = b$), the third is by the multiplication, and the fourth is again by absorption ($c(a+c) = c$)

0

Plug in some binary numbers or vectors to find out where the error occurs. E.g. $$a=(1,0), b=(0,1),c=(1,1)$$ then the first line evaluates to $$(a'b' + c)(a + b)(b' + a'c')'=(0,1)$$ and the last line to $$abc=(0,0)$$ So you can find the line where the error occurrs.