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.
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.