Hello everyone.
Welcome to this lecture on Boolean algebra.
This is a fundamental concept that powers modern computers, digital electronics and all logic based
systems.
This topic lies at the heart of computer organization, digital circuit design and decision making in
software logic.
Now let's begin with the deeper understanding of its origin and significance.
In the 19th century, an English mathematician named George Boole developed a new branch of mathematics
to model logical thinking using symbols and equations.
His central question was can human reasoning be expressed mathematically?
His answer formed the foundation of Boolean algebra, a system where variables only have two possible
values.
It's either true or false Represented true by one and false by zero in computing this binary system
load.
Logic problems to be expressed in a formal mathematical way.
The impact of this was.
Far reaching, laying the groundwork for logic control control systems.
Digital logic and eventually the structure of computer processors.
Now let's draw a clearer comparison here.
In traditional algebra, traditional algebra operates on real numbers like nine, 15, or 996, but
the Boolean algebra operates on a binary values.
It is either 0 or 1, right?
A or b.
The traditional gabra uses arithmetic operations like plus, minus and for example, modules and so
on and so forth.
But the Boolean algebra uses logical operations, for example, and or not.
And this goes on the third.
In traditional gabra we have infinite values allowed like you can combine, for example 999532 4 or
6 seven.
And this this is goes like this is infinite basically.
And in Boolean algebra you can use only two possible values.
It is either true or false.
In computer we call true one and false zero.
But for example in your own logic you can call the true t or f a or b.
Yeah, you get the logic here.
You can only have two values AWS, which can represent either true or false and result.
Result is what in the traditional algebra.
The result is a numerical computation and the Boolean algebra.
The result is logical decision.
As a beginner in Boolean logic, firstly we will learn about and or and not and returns one only when
both.
Both inputs are one.
So one and one equals 1 or 1 and zero equals zero.
It will only return Turn one.
If at least one or both inputs are one, but in or returns one if at least one input is one.
So if we have zero.
And one, we have one.
If we have zero and zero, we have zero.
And the node inverts the input, turning one into zero and zero into one.
So not one equals zero or not zero equals one.
One.
The Boolean algebra provides a compact and efficient way to represent truth conditions that occur frequently
in programming, hardware control, and logical automation.
So this was just the beginning to.
meet you with what you will learn.
In this section of our course, we will explain these conditions and values more deeply and detailed
in next lectures.
So don't worry.
Hello everyone, I'm Typhoon here again and in this lecture we are going to explore the basic boolean
operators the and or and not.
These are fundamental tools in digital logic and they are used in nearly every electronic device you
can think of, from computers and smartphones to microwaves and traffic lights.
If you're just getting started with a computer organization or digital systems, understanding these
operators is your first big step and I will guide you through each concept.
Explain how it works, show you the symbols, provide truth tables, and walk you through how this is
implemented in an actual electronic circuit.
So what is a Boolean operator A Boolean operator is a special kind of function that works on 1 or 2
binary values.
Binary values are simply 0 or 1, where zero usually means false or off, and one usually means true
or on.
Now the values that Boolean operators act on are called operands, and the result is also either 0 or
1.
Now Boolean operators are essentially act on are called operands.
So to make things clearer we will use truth tables.
A truth table is a tool used to show how the output of a logic operation changes based on all possible
combination of inputs.
But before we jump into the operators themselves, let's warm up with a real world example from binary
arithmetic.
So the third example we will do here is we will create x y and we will also have carry and we will have
some here.
So let's draw the lines here one.
And now we have the two binary values x and y and add them.
When you add binary numbers there are two things to think about.
The sum we have here.
This is the result of the addition and we have the carry here.
If both bits are one, we carry a value to the next position.
Now here's how the result looks in a truth table.
So here we will have zero and zero.
We have zero carry as well.
And the sum is what the zero we have 0101.
We have zero carry but the sum is one because of why is one here?
Or else we have one and zero again, same.
But if you have two ones, we also we will have another carry and the sum is going to zero.
But we will have just a small one here which we will not talk about that right now.
Now this is how binary addition works inside a digital circuit.
And now every row shows one possible input combination and what the circuit should output in that case.
And this as you can see, if we have both ones we will have the sum of zero.
Now we will return the binary addition later when we design circuits called adders.
But for now let's look at the core logic operations.
Logic gates are physical electronic components that perform Boolean operations.
Now think of them as a small building blocks inside a computer's processor.
Now each type of logic gate has a unique shape and behavior, and it performs one specific logic function
like and or not.
And in the next lecture, we'll cover each operator in detail, including its definition, the truth
table, and the circuit symbol.
But now we will start with the end.
So we have the and operator also called the contraction.
The type is binary operator.
All of these are binary operator.
How we can tell that it's a binary operator because it uses two inputs.
It is either 1 or 0.
So it's a binary operator.
And the symbols are symbols are this.
Now this is a common in logic and also common in programming.
Is this one just a dot.
Or you can also just write.
And this is a symbol symbols of ands here.
right.
The output there is rule is the output is one only if the.
Both inputs are one.
So let's do another example here.
We have a b value and we have x and or we have a and b.
So what we will have here we will just two three and four.
So if we have a zero and b zero we will get zero.
If we have a0 b1 we will get zero.
If we have a1 b0 we will get zero again.
But if we have A1B1 we will get one.
Now this is how and operator works.
So imagine two switches connected to the light.
The light turns on only if both switches are on or for example, like this.
This is a light bulb.
Here we have two switches and this is battery right.
This is just how and works.
And if you are designing computer circuits and has a circuit here.
So for example x and y the circuit here is what.
Now this is a circuit of and operated.
Now this is how you represent and in circuits here.
Now the gate is shaped like D as you can see here.
Now the behavior is similar to multiplication and regular math uh, which is uh one by one equals one.
Remember this is just like that.
And all other combinations are zero here as well.
But just this one.
And we also have the Or Operator, also called as disjunction.
The type is binary operator as well.
The symbols is V in logic, plus in math or just or in programming.
The rule is output is one if at least one input is one.
For example, we have a b and x or a or b, so if we have a zero and b zero, we will have what zero?
If we have A0B1 we will have one.
If we have a one and b zero we will have one.
If we have A1B1 we will have one as well.
Now you can see we got three and just one zero here.
Now think of the two doors that led to the same room.
If either door is open, you can enter.
Now that's how or works and how you represent or in logic is x, y, for example.
And this is how you represent or in, uh, circuits.
Now the gate has a curved shape.
Uh, it behaves like an addition.
Uh, as we mentioned here, uh, zero plus one is one, and one plus one is still one.
So we don't carry anything in the Boolean logic.
Now we have the not operator.
Now I would say that not operator is easiest among these.
Now the not operator, also called negation, is represented in a unary operator.
It uses only one input.
The symbols are.
This in logic and fs were like x or a line above the variable.
Now the rule is.
The rule is basically output the opposite of the input.
If A is one and not A is zero and vice versa.
So basically how we can write that is we will just have two.
So we have a here and we have not a.
This is so easy here.
So if A is zero not A will be one.
And if A is one then not A will be zero.
That's it.
The Not operator is like a mirror that always shows the opposite of what it sees.
If the input is true, the output is false.
If the input is false, the output is true.
The reversal process is known as inversion, and it's why the Not gate is also called an inverter.
Now let's use a practical analogy.
Imagine you are looking at a room with a smart light bulb.
The switch on the wall is programmed with a not logic.
If the switch detects a signal off, it turns the light on.
If the switch detects the signal on, it turns the light off.
Now, this behavior is especially useful in logic circuits when you need to deny or cancel a condition.
For example, a security system might activate an alarm only when a sensor signal is not triggered.
So in programming.
In many programming languages, the Not operation is written using this ternary symbol.
So um, basically not true equals false, and not false equals true.
And in Circuits.
The Not gate is drawn as a triangle like this.
For example a or let's actually represent it by x here triangle circle circle and what not x.
Now that is how you draw in a logic circuits.
The triangle represents the flow of logic and the circle means invert the input.
And now you will encounter not gates.
Frequently, when designing logic functions that require cancellation, inversion, or toggling a state
from on to off.
Now George Boole used multiple action for and and addition for or.
Why?
Because their behavior is similar to arithmetic, and using symbols made the system easier to work with.
This idea.
Connected logic to mathematics.
Allowing logic statements to be multiplied like equations.
Now that you know the basic logic operations, you will soon learn how to combine them to build more
advanced logic gate like with X or which is exclusive or true only when one input is true, not both.
You will learn that in next lectures and we have also Nand or non nor.
These are the combinations of basic operators.
Now you will also learn how these gates are wired together to create a circuit that can add, subtract,
compare and store information.
Now the way things your computer does every second here.
Take your time to review the truth tables and practice drawing the gate symbols.
This will help you build a strong foundation as we go further into digital logic.
Thank you for watching.
I'm Typhoon and I'm waiting you in the next lecture.