https:/.../emt171080p.mp4

Welcome to the next video on
the Boolean algebra and in

this video we're going to
look at simple truth tables.

Now before we start to look at
examples, I'd like to talk a

little bit about why do we use
the three tables and what are

these two tables. For now we use
the truth tables because these

are simple visual representation
of what an actual Boolean

expression can do. So for
example, you have a Boolean

expression combining together
two or three input values, and

then you want to know what are
all of those different

combinations. Expression can
take as an input combination and

you would want to know what is
the corresponding output for

every single one of those
combinations. So a Boolean

expression using the through
table can really easily show you

all of those different
combinations. Now, how do we

construct the truth table while
a through table which you have

seen in the previous video in
their most basic cases is just

showing you the input
combinations in the columns and

then. Breaking the complex
Boolean expression down into its

constituent parts and then
combining those parts together

according to different
operations that take place in

the actual expression. Now
depending on how many inputs we

have, you can use the following
rule to establish how many

combinations of inputs you can
have while every inputs can be

true or false or zero or one, so
that gives you.

Two possible input values. Now,
depending on how many different

input values you have, that will
be 2 to a certain power. So for

example, if I have got two
different input values, let's

say P&Q, then the possible
combination of these input

values will be two to two, which
is 4 if I have got three input

values, let's say PQ and.

Are then the possible
combination of all of these

input values will be two to
three, which is 8.

So how can we build up the start
of our three table for these two

combinations if I, let's say,
have two input values P&QI know

that I will have to end up with
four combinations of these. Now

everyone of these inputs can be
true or false. Now half of four

is 2. So if I place down to
trues two forces and then just

alternating tuzan forces.

Then I will have all the four
different combinations without

repeating any of them. Well,
it's my big too obvious for the

two combinations, but let's see
what happens if we have three

combinations. So let's
say if I had P.

Q. And R as my input
combinations, but half

of eight is 4, so I can
put down four truths.

And four forces.

Then I can put down to truths.

Two forces, two throughs to
forces Alternatingly and then

just one through 1/4 until I
fill up the whole table.

Now opposed to the two
combinations, which is

relatively simple because you
can just follow through OK. Both

of them can be true. Both of
them can be false, or 1314 when

it comes to three combinations
that starts to become a little

bit more complicated. So if you
don't remember this simple rule,

you might end up repeating one
or the other of the rows in this

combination, which would end up
you producing the wrong or

incomplete truth table for your.

Exercise and other important
things when it comes to these

three tables is to being able to
breakdown these different

expressions. So let's say that
you have got a complicated

expression P and not.

Q.

Or pee.

Now what's going on in here?

But I have got brackets just
like in BODMAS. I'm using the

bracket to rearrange the order
of operations, so the brackets

will Tammy that I need to
combine the inputs of P and not

Q before I can apply the OR
operation so that straight away

tells me that the priority to be
able to calculate things first

we will need to happen in the
bug inside the bracket. Now

inside the bracket I have got P
and not Q.

To be able to combine them
together, I will need to know

the values of P, which I can
just start from the beginning of

the table and then not Q. Now
where I'm building up the table,

I will have P&Q input values.
Therefore I will need to

calculate and not Q first to be
able to combine P and not Q

together. Once I've got not Q,
combined it together with the

using the end operation. Then I
can combine that.

Partial output with the OR and
the P. Now let's see how does it

work in practice.

So be.

Cute.

Not key.

B end not key.

And P and not Q or

P. So these are the different
input values that I need to be

able to calculate before I will
end up with the final answer, so

I've only got P&Q. I've only got
two different letters in my

expression, therefore I'm going
to use only two of those

combinations and the truth
values to start with are going

to be true to force force
through force 3/4. Now what does

the not due to an input?

Drew turns into force force,
turns into 2.

So that's what happens if I
apply the not operation to

the cube. What I need to do
next? I need to pick up the

note Q and I need to combine
that together with the P

using the end operation.

And This is why you need to
remember what every single one

of these different operations
do, because you need to be able

to apply them to certain
situations. So true and force

makes it force true and true
makes it true. Force and force

makes a force and force, and two
makes a force. So now I have got

answers to this operation for
each of the different

combinations. But that's not my
final goal, 'cause my final goal

is to have the answer for this
operation. So for that what I

need to do is pick up this.

Which is this spot?

And I need to use the
operation. Combine it

together with P again.

So what I'm going to do in this
case? I'm going to look at these

two. Input values and using the
rules of the OR operation true

or force gives me true true or
true gives me true force or

force gives me force force or
force gives me force.

What have I found?

What I found is that what is
this operation does to every

possible input combinations of
P&Q? What will be the output? So

if both of them are true?

The output will be true if P is
true, Q is forced, the output

will be true, but in the other
two cases, when PS4's and Q

history, the output overall
output will be for Stan. If both

of them are forced, the overall
output will be force. So with

this combination I have been
able to filter out these two

first input values. I'm letting
those go through this

combination. And these two I'm
going to stop. Remember that

these input values are always.
You can think about the movies

like electric circuits do I let
the signal go through or do I

not? And for some reasons it's
important for me to only allow

these two signals combinations
to and stop these two signals.

Let's look at another example.

P&Q

axor. P. Again, I've
only got two different letters

P&Q, so the overall different
inputs just P&Q, so I'm going to

have just a combination.

Do again the truth table will be
2 to force force or three

fourths 3/4 now and I'm looking
at the expression what are the

different inputs that I need to
combine together first before I

can calculate the overall
output. Now there is an X or

gate but there is a bracket
bracket is my priority just like

him but mass. So I need to
calculate the P&Q first before I

can apply. The P&QX or P so
the Exor gate will be the final

operation that I'm going to use
in here. So for P&Q simply.

Just use the end operation.

The end operation only returns
true if both of them are true.

In every other cases, the end
will be false.

And the X or what do I know
about the X or the X? Or is to

distinguish between the signals
being the same both through both

force or the signals being
different one through 1 force

and only letting through when
the signals are actually

different? That's why it's
called exclusive or it's

exclusively one or the other. So
what do I need to look? But the

first input will be the
combination that we just

calculated. And the other one is
the P so.

P and this combination is the
same, so the exclusivo returns

force. B and the combination is
different, so the exclusive or

gives you true.

These are again the same, so it
returns a force and these are

again the same, so it's again
returns of force.

So again, what have we found? We
know that P&Q can take all these

different input combinations and
the expression which is using

P&QX or P. We return these
output values. The interesting

thing about this expression is
it only let's through one

combination the true Force
combination. It stops every

other combination. So you can
see it. You can use these

combinations of different
operations of filtering.

Out certain inputs and only
allowing to go through one

type of input.

The next expression P or Q.

If then. Q.

So how many different letters do
I have? I only have P&Q, so this

through table again just gonna
have to input combinations.

The start will be again
true through Force Force

3, Four 3/4. Now try to
breakdown the expression.

There is a bracket again that
tells me the priority, so I will

need to find the P or Q first
before I can use.

D. Eve, then operation.

So B or Q both of them are true,
so this will be true. One of

them is true. This will be true.
Both of them are force. This

will be force. So that was very
simple. But how about if them

well? Just to make sure that
we remember what we've done

is, let's recall.

The truth table of the if then.

3344343
of course. Now what does the if

then do if both of them are
true? If then returns true if A

is true, but B is force if then
returns force if A is forced but

B is 2 if then returns true, and
if both of them are forced to if

then again returns true.

So this is one of the more
difficult tables to remember,

so I suggest that you learn
it very well so.

Back to our original table.

What are the two inputs that we
need to look into now? This one

is my A and this one is my
B. So if a is true and B

is true, the if then returns are
true if A is true, but B is

force that if done returns a

force. If both of them are
true, if then returns a true,

and if both of them are force,
which is the last guess in here

if then again returns true.

In the next video we will
look at few more simple

examples, but for now I have
a couple of practice

questions for you. You will
find the answers to these

questions at the end of the
video.

So these are the practice
questions.

And here are the answers.