OK we're gonna be looking at another example of K maps over here.
We have our p q and r and we have one two three four five ones so we'd really rather do this case because
we have three zeros that product for some sun and some products.
But here's the sum of the products we track we take we count each case looking for the ones in the sum
of products we have 0 0 1 is going to be not not yes because that 0 0 1.
The next case is a one.
The next phase of 0 1 is 0 1 0.
That's not yes not that's not yes not in central for each of the cases.
So this is our formula.
Now we're going to plug it in here.
We're going to do p q r right there.
That's the first case then P Q and Not R is there.
And we have peanut q and r is right there.
We have P with the Q not and the R is right here and we have P Q and are not is right there.
Now remember we can only do groups of 1 2 4 8 16 and we just have to cover them all.
So we could wrap around but.
And I'm going to do that just to show you so I can wrap around over here.
And so that's one group or two I could wrap around over here and that's another group of two.
And then I can wrap like this.
It's another group or two so you could have overlap but you have to only do you have to just make sure
you only covering it once.
Otherwise I don't have to do this because that's what it is.
So what.
So this first group that I did that wrapped around the PS canceled because of PMP not so I get a Q not
and I get an R and now the next case that I'm going to get is gonna be this which is going to P Q The
Peanut Q Because the RS cancel so I'm going to get peanut Q Because the RS cancel and then this case
over here the down I'm going to get the piece cancel so I get Q and Not so I get Q and are not.
So that's the that's the smallest version of this and we can come up with if we were to look for the
other ones if we look for the zeros then we have to flip it.
So that's good because at 0 0 0 0 and that's fine the next zero is ones yours a with the flip that one
so it's peanuts it's peanut Q R and this 0 is not not not as P not now in this over here again just
need to find each one that's a zero so the P Q R P Q R is right there the peanut Q Plus the honors right
here and then the peanut Q are not peanut not Ana is over here so that peanut unit are not I can't really
do anything with because it doesn't hit any group but this one wraps around to this group.
So this group over here is going to be the PS cam so we're left with the Q Plus her
I'm left with that and then I have this one which is not connected of anything which is P plus q not
plus Ana so that one doesn't go anywhere but that's going to be these two want these two things are
equivalent that would be perfectly fine.
So over here have a bunch of ones and zeros in this why am I going to fill it into where it belongs.
So let me fill in the ones this one right here is Peanut Q not r ze peanut Q not R C so that's right
there.
This next one it's kind of hard to keep track of it but this next one is Peanut Q are not Zina so peanut
Q are not Zina it's right there and this next one is Peanut Q are not ze peanut Q are not ze That's
right there OK.
And then this next one is Peanut Q Are Zena so peanut Q Are Zena that's that one this one over here
is p q nut are not ze so p q nut are not ze is right there this one over here is p q not r ze not P
Q nut R xenon right there and this one is p q are not Zina p q are not Zina sorry about that and this
one is p q are not c p q are not ze and this one over here is p q are the not P Q R Xena.
So now we have to do the groups that are of 2 1 to 40 now theoretically you might say you can do this
group and go across like that the only thing is that that doesn't count because that's not a group that's
a group of six not a group.
So the largest group of four that's together is this then there's not much else I can really do to make
this any smaller I can do this group too they're just fine I can do this group it too and this group
it 2 in this group 2 for instance.
So it's going to get a little messy but I have this group of 2 in this group of 2 I could have done
that.
That seems like the easiest thing to do and then this guy is gonna be all by itself.
So for each group we have to write out what we have.
So this group before the PS go away we have a Q The R is there.
So we're left with P and the PS go I'm sorry.
So that's a Q rather let me rephrase that for you so it's clear.
So we're left with the.
From this the PS go we left of the Q from the group before and the R's are still there but this goes
away and so that's the first thing from the group before from this group to we're left with a P the
Q S go away and left the RNC are not Z OK.
Let's see what else we're left with we're left with this whole group.
So that's peanut Kuna R ze peanut Kuna ANSI sorry for the messy handwriting here.
And for this to group right here we're left with Q Are they not.
So that's a plus.
Q Or is it not.
And then my last group I'm going to add in is this group right here which the PS the QS go way is P
R is not.
So that's P R ze nodded.
So that's our group of four over there and you can get this group before in a couple groups of two in
group one not very nice you know kind of a little bit ugly but we were able to get everybody in and
did every case as we suggested.
So that's another example of the.
Looking for the sum of products and then this is the product of the sums that are here so you're looking
for the opposite and when you're looking at the sum of the products looking for the ones some of us
are looking for the zeros.