I need to simplify the following boolean expression
¬A¬B¬C + (B ⊕ C) + A¬B
I know B⊕C = ¬BC + B¬C
Then the expression will become
¬A¬B¬C +(¬BC + B¬C) +A¬B
However, I'm stuck on it and I don't know how to simplify it further. Can someone give me a hint or push me in the right direction? Thanks in advance
Best Answer
A couple of useful principles are:
Adjacency
$AB+AB'=A$ (I find the ' a little easier to work with than $\neg$)
Absorption
$A + AB = A$
Idempotence
$A + A = A$
So, starting with:
$A'B'C'+B'C+BC'+AB'$
(use Adjacency to rewrite $AB'$ as $AB'C+AB'C'$)
$= A'B'C'+B'C+BC'+AB'C+AB'C'$
($B'C$ absorbs $AB'C$)
$= A'B'C'+B'C+BC'+AB'C'$
(use Adjacency to rewrite $A'B'C'+AB'C'$ as $B'C'$)
$= B'C'+B'C+BC'$
(use Idempotence to make a copy of $B'C'$)
$= B'C'+B'C+B'C'+BC'$
(Use Adjancency to rewrite $B'C'+B'C$ as $B'$ and $B'C'+BC'$ as $C'$)
$= B'+C'$