# Boolean expression simplification using 3 variables

(!A B !C) + (B !C !D) + (!ACD) + (!BCD) + (!ABD)

= (!A B !C) + (B !C !D) + (!ACD) + (!BCD) + (!ABD)(1)

= (!A B !C) + (B !C !D) + (!ACD) + (!BCD) + (!ABD)(C + !C)

= (!A B !C) + (B !C !D) + (!ACD) + (!BCD) + (!ABDC) + (!ABD !C)

= (!A B !C) + (B !C !D) + (!ACD) + (!BCD) + (!ABDC) + (!ABD !C)

= (!A B !C) + (B !C !D) + (!ACD) + (!BCD) + (!ABCD) + (!AB !CD)

= (!A B !C) + (!AB !CD) + (B !C !D) + (!ACD) + (!ACDB) + (!BCD)

= [(!A B !C)(1 + D)] + (B !C !D) + [(!ACD)(1 + B)] + (!BCD)

= (!A B !C) + (B !C !D) + (!ACD) + (!BCD)

At this point I don't know how to continue the problem. I verified it using a calculator and seems there is still some simplification to do.

Here are two handy equivalences:

$$PQ + P!Q=P$$

Absorption

$$P + PQ=P$$

With that:

$$(!A B !C) + (B !C !D) + (!ACD) + (!BCD) + (!ABD)\overset{Adjacency \ x \ 2}=$$

$$(!A B !C D) + (!A B !C !D) + (B !C !D) + (!ABCD) + (!A!BCD)+ (!BCD) + (!ABD)\overset{Absorption \ x \ 4}=$$

$$(B !C !D) + (!BCD) + (!ABD)$$

I am sure that went a little quick though, so here is an explanation:

In step 1, I applied Adjacency to:

$$(!A B !C)$$ ... becoming $$(!A B !C D) + (!A B !C !D)$$

and to $$(!ACD)$$ ... becoming $$(!ABCD) + (!A!BCD)$$

Then, in step 2:

$$(!A B !C D)$$ got absorbed by $$(!ABD)$$

$$(!A B !C !D)$$ got absorbed by $$(B !C !D)$$

$$(!ABCD)$$ got absorbed by $$(!ABD)$$

$$(!A!BCD)$$ got absorbed by $$(!BCD)$$

More equivalences to put in your toolbox!

Now, notice that you effectively did Adjacency as well in your first few steps, expanding $$(!ABD)$$ to $$(!ABCD)+(!AB!CD)$$ ... and yet that one isn't as useful as the ones I did. So, I know you are going to ask: how did I know which terms to expand using Adjacency? Well, I created a little K-map. Look it up!

• Oh I finally understood what Adjacency is. Jesus Christ great work. Thank you for taking all that time for this as well as the K-map. On a side note, do you know of any good Boolean algebra calculators. I've tried plugging the problem in but it seems it gives me a different answer compared to yours. – JohnJohnyPapaJohn Feb 19 at 2:44
• @JohnJohnyPapaJohn Oh? What answer did your calculator give? Oh, and yeah, this form of Adjacency is a lot more intuitive than its dual I used in the other question (the $(P + Q)(P + !Q)=P$ one). Indeed, the $PQ+P!Q$ form is what you use in K-Maps – Bram28 Feb 19 at 3:06
• @JohnJohnyPapaJohn wolframalpha.com/input/… It shows the same DNF I arrived at .. but shows that the CNF is even more efficient yet. Looking back at the K-Map, I see that now as well. I missed that :( – Bram28 Feb 19 at 3:14