Okay, Welcome back! In this video we will use Boolean Algebra to simplify a few
expressions. We will work through a couple of examples and I'll highlight which
rules are being used. So just a quick summary of what rules we're gonna use. So
here are some Boolean Algebra rules. These rules are based on the AND gate
logic so here are my AND gate logic rules and here are my OR gate logic
rules; so these rules are fairly simple! There are a few additional rules that
that are useful. This rule is also sometimes called the
Absorption Rule -- which states A OR (not A).B is equal to (A or B) and then the
following two rules are called De Morgan's theorem. It basically says that
A or B not so a nor B is the same as saying not A ANDed with not B and this
one says A NAND B is the same as saying not A or not B so our goal in this
exercise is not to prove these rules but to use these rules to simplify some
expression so let's do a couple of examples so here's my first example f
equals ABC or not A or a not B C so let's take a look at this and we see so
how do you proceed here so I see and the first term and the third term there's a
and C common to both of these guys so let's bring this up on this side and
take easy out so we have a and C from this B up here we have a and C and what
we have left is not B and then We will just leave not A alone right here for
now now if you take a look closely up here B or not P so we have a or rule
that says a or not A is equal to one so be or not be
now simplifies itself to become a 1 so that's what we have here so we have a C
ANDed with 1 or not A if you look at and set of rules we have
A + 1 is equal to A. So here we have a C ANDed with a 1 so that should give us
just C or not A . So we have C or not A. Now this is where to requires a little
more practice to see this but this is basically this absorption rule take a
look at this F equals not A or a C now if I have a and I send it through a not
gate I get not A if I take a not A and I send
that through another not gate I get a back but I actually get not of not A so
is the same as a with a double bar over it
so if you look at this expression right here I have not A and a with a double
bar over it so this looks like not A or not of not A and it with C that is very
similar to this rule right here absorption rule so you take a look
carefully right here so we ended end up with not A or C so that's the final
simplified expression for the original expression we started out with okay all
right let's do two more examples so in our second example we have f is not A
and it with not B and it would not C or not A not BC or not C. Now what
should we do first here it's fairly obvious at the first in the second term
have a lot of things in common first in the second term both have not A not B so
let's bring not A not B out and what we end up with is not A not B ANDed with
not C from here and see from there or with not A and it would not see so We will
just leave it this term as is now again using the same
rules of or not C or C will go into a1 and again using the same rule I ended
with one in this case not A not B ANDed with 1 is simply 1 so what we end up
with is f it was not A not B or not A nazi can we simplify this even further
now mathematically this is the most simplified Boolean expression we could
however take not A out of this this is not technically simplification because
we haven't really gotten rid of any variable when we build a circuit this
particular expression might lead to a different circuit than this but this is
not really simplified from a mathematical standpoint okay but so we
could leave this as the final answer or we could leave this as the final answer
now some of you might see that not B or not C is this team organ theorem right
there so you might want to write this as yes so this is yet another correct
answer so there's no no real simplification happening in these three
steps it's just a different way of writing the exact same expression now
some of you might see that this is not A ANDed with let's call this BC as P so
not A ANDed with not P well here's another team Oregon law that we might be
able to use so not A ANDed with so if I call this P so not A ANDed with not P is
the same as a or P with a bar over it or a nor P so we could write
this as a or PC and a bar over it okay so any four of these expressions in
purple are valid and correct simplified Boolean expression for this originalist
expression we started out with all of these are correct okay let's do one more
exercise in this final example We will take a look at the following Boolean
expression it looks just a little more complicated
than the one we started with but it's not so here you go F is equal to a and
it would not be ended with in parentheses we have C or B D or not A
not B and this whole thing is being ended with C so let's take a look so
what we do here so let's do first thing we let's let's distribute a not be
inside here so as such so we have a not be and it would see from this term here
we have a not B ANDed with B and it D from that term will leave this term
alone and We will leave the final and it's the outside as is now if you take a
closer look here I have B not B what's B and it would not be P ended with the a
ANDed with not A is equal to zero so B and it would not be it leads me to a
zero now if I look at the end rules again zero ended with anything is zero
so we have another end rule that we can use so this whole term will go to a zero
thus leading us to two simply a not BC that was the first term from here not A
not B and finally and it would see let's take a look here what can we do next
well let's bring in C inside so let's distribute this C so I have a not
be not see that was the first term and it would see or not A not B ANDed with
my C so what can we do now so let's take a look here I have to seize a ANDed with
a is a so C ANDed with C is simply C so I have a and it would not be C or not A
not B C what can we take common here so I see not B ANDed with C not B ANDed
with C let's take that out and We will have a from the first term and not A
from the second term or together so a or with not A and that whole thing is and
it would not be not C so a or not A that rule right there is equal to one so I
have one ANDed with B C and we are left just with not B C so what we looked we
use all four of the end rules in this particular example so this Boolean
expression we started out with simplify simply to not be C