SOP expression simplify

Question

I wanted to know how to simply this SOP expression using Boolean Algebra :

F= A'BC'+A'BC+ABC

I got this answer using K - Map: (A'B+BC), but I want to know how do get it using Boolean Algebra Rules.

Thank you.

Answer 1

You can relate the Boolean Algebra manipulations directly to what you do in a K-map.

Consider the K-map for this expression:

Now, note that the red oval represents the expression $A'B$ while the blue oval represents $BC$. Moreover, the red oval covers the terms $A'BC$ and $A'BC'$, while the blue oval covers the terms $ABC$ and $A'BC$.

So, we have that $A'B = A'BC + A'BC'$ and $BC=ABC+A'BC$, which are both instances of the general Adjacency principle:

Adjacency

$P = PQ + PQ'$

Indeed, note that Adjacency is named exactly after the fact that you combine two adjacent terms in a K-Map into one.

OK, so we thus have that:

$A'BC+A'BC'+ABC+A'BC \overset{Adjacency \ x \ 2}= A'B+BC$$

Hmmm ... that is almost your starting expression $A'BC'+A'BC+ABC$, except there are two $A'BC$ terms instead of one. However, we can get that second one easily enough:

Idfempotence

$P = P + P$

OK, so we are now ready to apply the Boolean algebra on your original expression:

$A'BC'+A'BC+ABC \overset{Idempotence}= A'BC'+A'BC+A'BC+ABC \overset{Adjacency \ x \ 2}= A'B+BC$$

Answer 2

I hope this is more easy for begginers !