Hello and welcome to the first video on Boolean logic.
And Boolean expressions. You might know this topic in an
alternative name because the same topic sometimes also
referred to as digital Logic or Boolean algebra or algebra of
proposition. What this topic is looking at is looking at a
collection of input values.
A collection of operators.
That sometimes we refer to as logic gates and looking at what
happens at the end, what is a particular set of of these
inputs and operators? What kind of outputs you can end up with?
Now the origonal part of this coming from the at the very very
beginning, the computer has been built up from logic circuits.
Tiny, tiny little circuits and then depending on which circuits
you connected together, you were able to tie the computer to do
different things so.
The input and output values can be recorded in a number of
different ways. Again, there are two common ways.
One is the true for set up and the other one is the 1 zero set
up again. It comes from the fact that logic circuits can be
turned on or off true and one is equal to the on position and
force in OR equal to the off position. Remember circuits turn
it on or turn it off. So let's
look at. What are the basic Boolean operations? The
first Boolean operation?
Is the end operation now this end operation sometimes just
written out as end in capital letters sometimes is referred to
as this symbol?
And sometimes it's referred to as a multiplication simple
symbol because it works like ordinary multiplication within
the binary system. Now it also has got in digital logic. If
you're looking at the circuits themselves. If you're looking
at how to build boards together, it has got a logic
gate symbol and that logic gate symbol is this one.
Now this tells you that this endgate can take a minimum
of two input values and it will give you always one
output value.
So what does the end gate do now? The end gate is actually
working as if you had a water pipe fixed with two taps on it,
one after the other.
So when we have water flow through, the water will flow
through only if both of the tabs are open. If I turn
this tab off.
The voter will stop here if I turn this step on when I leave
this tab of the water will stop here, so water still doesn't
flow through an. Obviously, if I have both taps turned off,
the water doesn't go further than here. So what are we only
go through if all of these steps are on, so would have can
go through only this way. Now how can we sort of summarize
this information in a nice and simple visual format? That's
why truth tables.
Has been invented through. Tables are simple tables which
tells you what are the input values or other possible input
values that can come. What is the logic gate or what is the
Boolean operation that we use here and then? What will be the
end result of that Boolean operation for every possible
income combinations? Now let's look at the simple examples.
Let's see how does the P&Q
operation works. B is an input queue, is a different input.
What happens if I combine them together? What will be the
common output so P.
And Q.
And here will be my output.
Now what kind of setups can the two taps have? While I can turn
both of them on?
I can turn one of them on the other one off, all in
the other combination.
Or I can have both of them turned off.
So what did we say? The water can only go through if both of
the taps are on?
Every other combination will stop the water from flowing, so
this is the truth table accompanying the end.
Operation. With this Boolean trip tables, you need to know
them. You need to understand them because later on we'll be
combining more than just one single operation together and
see what happens if we start to mix them up. So this was the
first one, the end operation. Let's have a look at the next
one, which is the OR operator. Now the symbol for the OR
operator can be this small away. The opposite, the turned upside
of the end or the addition because it works.
Like the audition.
And if you are coming from the engineering background.
You can see either this.
Symbol or this symbol?
For the OR gates, again it takes in at least two
incoming values and gives you one outgoing value, so at
least two inputs, one output.
You can think about the OR gate.
As water pipes but fixed
now. In a different way now these water pipes I fixed
together in a parallel fashion and on each branch
we have got a tap.
So what happens in this case? Now if I turn this tap off, stop
the water flowing here, but I don't turn this stuff off.
The water will be able to bypass that turned side and fluid flow
through. Here the same the other way around. And obviously if
both of the tabs are open then the voter have got the choice of
flowing through one or the other, so the OR operation.
Opposite to what the end does, it only stops the water. In one
case it stops the water if both of the taps are turned off.
So what does it look like in the truth table fashion? So again.
P or Q. What are the possible income combinations and what are
the possible outcome combinations of these? So again,
I can have two values.
Two input values P.
K.
So again, what are the different input combinations for these two
values? The P&QA quick trick.
True, true Force Force 3434. This fact comes from I've
got two input values.
P or K, but all both of them can be true or false, so I only have
got two possible switch stands. So 2 to the power of two gives
me 4, but two is the number of input values and two is the
number of possible outcomes like trues or force.
So what did we establish if both of the taps are turned
on, then the voter can flow through. If one of the taps
are turned on, then the water can flow through that branch
and bypass the off tab.
But if both of the taps are turned off than the water it
stopped. So that's when this this operation is forced.
So the next operation that I would like to talk about is the
not operation, which sometimes also used this symbol.
Sometimes this symbol and sometimes it used just by a bar
over the input for the pictorial symbol for the Northgate. For
engineers is this it's a little triangle with a little circle at
the end now.
Compared with the others, the not operation only have one
input and has one output.
OK, so that state to wait as you something. So if I have got just
one input then that input P.
Can only be true or false and then not P.
But what is not true? What is not true is force and
what is not force is true. So the not operation has
got a very special role. It flips it inverts it changes
the input to its opposite.
The next simple operation is the X or.
Which has got this symbol, so it doesn't have that many symbol as
the not. So that's easier. And the exors pictorial symbol for
joining as circuit.
Is the OR gate?
But with an extra leg added to it so it can take again at least
two inputs. Or if we use the alternative way of the X or.
And it would look.
Something like that.
OK, now this is called the axle operation because it's
exclusive.
Or
so it's exclusively one or the other input, so the xclusive
or the X or operation filters out the input values when the
inputs are the same. So what do I mean by that? If I have
got inputs P&Q?
What will the acts or do to them?
So again, be can be true to force force and Q is
true force three force.
So exclusive, or if the inputs are the same, which is this
case? Because both of them are true. The Exor Gate gives you a
4th signal. Basically the axle gates stops the signal going
through. If one is to the other one is forced then that's when
the signal can go through and again because force enforces the
same input value that the EXOR gate stops your signal going
through. And remember that I mentioned at the beginning of
this video. These are based on the electric circuits and then
you wanted to manipulate at the very, very early stages or
lippit stages of computing you wanted to manipulate where the
electrical signal goes doesn't go through here doesn't go
through that you wanted to manipulate and filter out
certain inputs in favor of other inputs. So these different gates
give you that kind of option of turning them around. Saying, I
don't want this input, I want that that input combination to
go through and nothing else.
There are a couple of more operations that I would like
to talk about. These are slightly more complicated. I
can't really give you any nice and simple example of why they
work in here. We just have to learn that this is the way
that they work, so one of them is that you've done.
And the symbol for that is this forward error.
So again. I've got inputs P&Q.
And then what will be the Alpha output of the
P IF then Q operation?
So true true Force force three force, three force.
Sometimes this also called the implies.
So true implies true, that is
true. But true cannot imply force, so this one is force
force can imply true.
And force can imply force that's true. Again, this is
probably going to be the most difficult gate to understand
why this works. You just have to learn the truth tables, and
once you know the truth tables, you can apply it to
any kind of combinations.
And the last operation that I would like to talk about is that
if and only if.
And the symbol for that is an arrow that goes both
ways. So if I have got the two inputs again P&Q.
The P if and only Q will work this way.
33443434.
This one is only true.
If both inputs are the same, so true and true are the same, so
this will be true. True annefors are different, so the output
will be force same for the third right and for some for side the
same. So this is true in here. Now if you look at this one and
if you remember the X or you can support that these two
operations if and only if an the X or are doing exactly the
opposite in this using this.
Operation I can filter out the same input values and stop the
different input values.
So that's again a very useful operation to have.
This short video was intended to expose you to the basics of the
Boolean logic or digital logic and show you had the truth.
Tables can be built up and what are the most commonly used
operations to be able to follow up on digital logic? You will
need to be able to know this by heart, so these are different
operations that every time you need to apply them you will be
have to be very, very confident knowing these operations how
they work. What they do? What kind of inputs they let
through an? What kind? What kind of inputs they stop from
going through? And again, as I mentioned at the beginning,
this is all coming from the basic principles that went
first. Human started team when the computers they build them
together from very tiny basic circuits.