Welcome to the last video. In
the Boolean series. In this

video we will look at some of
the more complicated Boolean

expressions and therethrough
tables.

Now let's make matters a
little bit more complicated.

By using more brackets.

P&R

or
Q&R.

If then. P or not Q first
thing that we want to establish

is to find out how many
different inputs are there.

BRQRPQ there are
altogether three different

letters, so P.

Q.

R.

Again, three different letters.
Two to three gives me 8, so I'm

going to have eight different
input combinations or eight

different rows in the truth
table. Half of it is 4, so I'm

going to start with four truths
and then four forces.

Then half of four is 2, so I'm
going to have two 2, two fours

until I fill up the table.

Then just use true force.

The force to fill
in the last column.

Now let's look at the different
brackets. I have got a square

bracket here. Which is combining
together this bracket using the

event operation. So to be able
to use the if then operation

first I need to know what is the
value inside the square bracket,

but unfortunately inside the
square bracket I have got two

different brackets. So first I
need to calculate the P&R.

Then I need to calculate
the Q&R.

Then I will be able to calculate
what are the values in the

square bracket, not to save some
space. I'm just going to put the

square brackets sign into here.

Now once I've got the square
bracket, I need to combine the

values of the square bracket
with this bracket, but inside

this bracket I have got an not Q
as well. So first I will need to

calculate the value of not Q,
then I will be able to calculate

the P or not Q values and once
I've got that I will be able to

use the value for the event
operation. And because there are

no other if their operations.

I can use the even operation
here instead of having to write

the whole thing out because then
that would again just take up

too much space in here.

So let's get started P&R I'm
using these two columns, the P

and they are columns and I'm
using an end again. What do I

know about the end and is only
true if both of them are true?

This is true for both of them.
That is not so it will be

forced. That's true for both of
them. That's going to give me

true that is not gives me a
force that is not give me a

force force force and force

again. Then I'm going to have to
use the Q&R so Q&R both of them

are true, gives me back a true
value. They are different. That

gives me back of force value.

They bought force that still
gives me a force value because

it's the end. Both through gives
me true Force force force. Now

that's great. What I need to do
next is to find out what is the

overall value combinations in
the square bracket. Now in this

work square bracket I'm
combining together these two

using the OR. So this is this
column and that is this column

and I'm going to need to combine
them together using the OR. What

do I know about the?

Or the OR is only force if both
of them are forced in every

other case, is the OR returns
true? So that's true. This is

force. That is true. That
is force true Force, force

force. So I've got the
square bracket.

Next thing I need is
the note QQ is here.

So they're not. Q would be
just where Q is true, not

QB LB force.

An when Q is force not not Q
will be true.

I've got the not key now and
what I need is the P or not Q.

P or not, QP is this one.

And not Q is this one that they
are quite far away from each

other, but it just an or
operation. Again, if at least

one of them is true, then the
orbital return are true. So true

force gives me true. True force
gives me true true. True gives

me 222 gives me true force and
force gives me a force force and

force give me a force for so
true gives me a true and false

or true. Me too again.

And then the last step.

Is that I need to use?

The If then operation.

Which two columns do I need to
use the Ethan operation for? I

need to use for this?

And I need to use for the
square bracket. Now this is

my a because this is what on
the left side for the event

symbol and this is my B
because this is on the right

side of the if then symbol.

True if then true returns a true

value. Force even true returns a
true value true if then two

returns at true Value Force if
then true returns with true

value. True if then force
returns a force, why you force

if then forced returns a true
value force if then true returns

a true value, and force Tipton
true again returns 2 value.

There is a curiosity about this
to table because if you look at

the final output values, there
is only 1/4 output in here.

Everything else is true.

Let's finish this video with a
very complicated truth value

table. Let me show you what you
have to do. If you have gold 4

input values, so.

PX or Q.

And not R.

If and only if.

Not Q or S.

Now I have got four different
letters PQ R&S not 2 to 4 =

16. So I'm going to have a
rather long truth value table.

So let's start building it up.
I've got P.

I've got Q.

I've got our.

And I've got ***.

Now what's the half of 16
half of 16 is 8, so I'm

going to have to start with
putting down 8 truth.

And eight forces.

Just about then the
next step I'm going to

put down four truths.

N 4 forces.

4 two again.

And for force again.

Next, 1/2 of four is 2, so
two through 2 fours.

To to force force again until
I finish with the whole table.

And in the last one is just 3/4,
alternating all the way down.

OK, does this start up the
table? Then what I need to

look at is breaking up the
brackets, finding out the

priorities so there is a
square bracket and within the

square bracket I have got a
normal bracket and are not so

I can cut combine together PX
or Q.

Then I would need to
find out the note R.

In the five, being able to
calculate that, I will be

able to calculate what is
inside the square bracket.

Then once the square bracket has
been done, then I need to look

at what's on the other side of
the if and only if symbol on the

other side. I have got not Q or
S to be able to calculate that.

First I need to calculate not Q.

Then I can do the not Q or S.

An once that's been done, I can
calculate the if and only if,

which will give me the final
output. So let's get started PX

or Q. What do I know about the X
or the X Rays, the xclusive, or

which is only true if one or the
other is exclusively true? So

basically, if the inputs are
different than the X, or will be

true PX or Q?

So the first 2 columns
they are the same, so the

actual returns are force.

They are different. The EXO
returns are true.

They are different, so the
exhaust air returns are true.

And in here they are the same,
so the X or returns are force.

So the next column in R2 value
table will be the not R. So

where are is true, North Star
will be force an when R is

force, not R will be true.

Force Force 3
Three Force Force

3. Two force for street due
again, so I've got my starting

values again. I need to
calculate what's inside the

square bracket, so what's inside
the square bracket I'm combining

together the X or and the North
R using the end operation.

What do you remember about
the end operation? It's only

true if both of them are
true, so let's came through.

That is, both of them are true,
so I'm going to place a true

sign there, same here.

Here.

Here.

And the rest of them are all
different or double forces, so I

can just fill in the last of the
table with force.

Then calculate not Q while Q is
here and not Q will be just the

opposite of what Cindy Q column
so true will be force.

Annefors will be true.

And then what I need to do here
is combined together, not Q.

An S.

Using the OR operation. So
what do I remember about the

OR the OR is only force if
both of them are force? So

let's try to filter out
those inputs, not both.

Force both of them are
force, so this will be a

force in here.

Both of them are force,
so that's another force.

Both of them are force.
Here is another force.

Here is one more.

So the rest of them I
can fill in with truth.

Lovely, so the final answer is
the if and only if and what do I

combine together to get the
final answer? On one side I've

got the square bracket which is
here and the other side I have

called the note Q or S which is
here. So if and only if returns

true. If the inputs are the same
and it returns force if the

inputs are different.

To enforce difference, so the if
and only if gives me force force

and force the same. So the if
and only if. It gives me true.

Different force, same true,
different force, different

force, same again true,
same different same.

Same.

Different gives me a force
different gives me a force

different, gives me a force,
different, gives me a force and

a different at the end. Again
gives me or force. So this is

the final answer for this long
and rather complicated Boolean

expression. And as you see there
are quite a bit of different

combinations of force and truth.

I hope that by now you have
been able to build a good

understanding of the Boolean
expression and they threw

tables. I have a few last
practice questions for you

and the answers will follow,
so these are the practice

questions.

And here are the answers.