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# Boolean Algebra And Digital Logic Pdf

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Published: 25.04.2021  Boolean algebra can be considered as an algebra that deals with binary variables and logic operations. Boolean algebraic variables are designated by letters such as A, B, x, and y. The Boolean algebraic functions are mostly expressed with binary variables, logic operation symbols, parentheses, and equal sign.

## Boolean Algebra

We all love computers. They can do so many amazing things. Within a couple of decades computers have completely revolutionized almost all the aspects of human life. They can do tasks of varying degrees of sophistication, all by just flipping zeros and ones. It is remarkable to see how such a simple action can lead to so much complexity.

But I'm sure you all know that such complexity cannot be achieved practically by just randomly flipping the numbers. There is indeed some reasoning behind it. There are rules that govern the way this should be done. In this article we will discuss those rules and we will see how they govern the way computers "think". In his book, British Mathematician George Boole proposed a systematic set of rules for manipulation of Truth Values.

These rules gave a mathematical foundation for dealing with logical propositions. These sets of foundations led to the development of Boolean Algebra. To best understand Boolean Algebra, we first have to understand the similarities and differences between Boolean Algebra and other forms of Algebra. Algebra, in general, deals with the study of mathematical symbols and the operations that can be performed on these symbols.

These symbols do not have a meaning of their own. They represent some other quantity. It is this quantity that gives some value to these symbols and it is this quantity on which the operations are actually being performed. Boolean Algebra also deals with symbols and the rules that govern the operations on these symbols but the difference lies in what these symbols represent.

In case of ordinary Algebra, the symbols represent the Real numbers whereas in Boolean Algebra they represent the Truth values. The image below shows the entire set of Real numbers. The set of Real numbers includes Natural numbers 1, 2, 3, Ordinary Algebra deals with this entire set of numbers. The Truth values, in comparison, consist of a set of only two values: False and True. Here, I would like to point out the fact that we can use any other symbol to represent these values.

For example in Computer Science we mostly represent these values using 0 and 1. You can also do it in more fancy ways by representing truth values with some other symbols such as Cats and Dogs or Bananas and Oranges. The point here is that the internal meaning of these symbols will remain the same irrespective of the symbol you use. But make sure that you don't change the symbols while performing the operations.

Now the question is that if True and False , 0 and 1 are just the representations, then what is it that they are trying to represent? The underlying meaning behind truth values comes from field of Logic where truth values are used to tell if a proposition is "True" or "False". If the above proposition is true then we assign it the truth value of "True" or "1" otherwise we assign it "False" or "0".

In Digital Electronics, truth values are used to represent the "On" and "Off" states of electronic circuits. We will discuss more about that later in this article. Just like Ordinary Algebra, Boolean Algebra also has operations which can be applied on the values to get some results. Although these operations are not similar to ones in ordinary algebra because, as we discussed earlier, Boolean algebra works on Truth values rather than Real Numbers.

OR : Also known as Disjunction. This operation is performed on two Boolean variables. The output of the OR operation will be 0 when both of the operands are 0, otherwise it will be 1. To get a clearer picture of what this operation does we can visualize it with the help of a Truth Table below. AND : Also known as Conjunction. The output of AND operations will be 1 when both operands are 1, otherwise it will be 0.

The truth table representation is as follows. NOT : Also known as Negation. This operation is performed only on one variable. If the value of the variable is 1 then this operation simply converts it into 0 and if the value of the variable is 0, then it converts it into 1. After its initial development, Boolean Algebra, for a very long time, remained one of those concepts in Mathematics which did not have any significant practical applications.

In the s, Claude Shannon, an American Mathematician, realised that Boolean Algebra could be used in circuits where the binary variables could represent the "low" and "high" voltage signals or "on" and "off" states.

This simple idea of making circuits with the help of Boolean Algebra led to the development of Digital Electronics which contributed heavily in the development of circuits for computers.

Logic Gates are the circuits which represent a boolean operation. For example an OR gate will represent an OR operation. Alongside the basic logic gates we also have logic gates that can be created using the combination of the basic logic gates. NAND gate gives an output of 0 if both inputs are 1, otherwise 1.

NOR gate gives an output of 1 if both inputs are 0, otherwise 0. Most digital circuits are built using NAND or NOR gates because of their functional completeness property and also because they are easy to fabricate. Other than the above mentioned gates we also have some special kind of gates which serve some specific purpose. These are as follows:. XOR : XOR gate or Exclusive-OR gate is a special type of logic gate which gives 0 as output if both of the inputs are either 0 or 1, otherwise it gives 1.

So, with all that we can now conclude our discussion on Boolean Algebra here. I hope by now you have a decent picture of what Boolean Algebra is all about.

This is definitely not all you need to know about Boolean Algebra. Boolean Algebra has a lot of concepts and details that we were not able to discuss in this article. If you read this far, tweet to the author to show them you care.

Tweet a thanks. Learn to code for free. Get started. Forum Donate. Aditya Dehal. What is Boolean Algebra? The rules I mentioned above are described by a field of Mathematics called Boolean Algebra. Aditya Dehal Read more posts by this author. ## Boolean Algebra Truth Table Tutorial – XOR, NOR, and Logic Symbols Explained

Learning to analyze digital circuits requires much study and practice. Typically, students practice by working through lots of sample problems and checking their answers against those provided by the textbook or the instructor. While this is good, there is a much better way. For successful circuit-building exercises, follow these steps:. Always be sure that the power supply voltage levels are within specification for the logic circuits you plan to use. If TTL, the power supply must be a 5-volt regulated supply, adjusted to a value as close to 5. One way you can save time and reduce the possibility of error is to begin with a very simple circuit and incrementally add components to increase its complexity after each analysis, rather than building a whole new circuit for each practice problem.

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This is the digital electronics questions and answers section on "Boolean Algebra and Logic Simplification" with explanation for various interview, competitive examination and entrance test. We have also provided number of questions asked since and average weightage for each subject. Again, the simpler the Boolean expression the simpler the resultingthe Boolean expression, the simpler the resulting logic. Table Basic rules of Boolean algebra. Worksheet 1. Intermediate Algebra Worksheets.

It pro- vides minimal coverage of Boolean algebra and this algebra's relationship to logic gates and basic digital circuits. You may already be.

## ❞ كتاب 02 – Boolean Algebra and Logic Gates ❝ ⏤ إم موريس مانو

Morris Mano Chapter 3 Gate-level minimization refers to che design task of finding an optimal gate-level implementation of the Boolean functions describing a digital circuit. This task is well understood, but is difficult to execute by manual methods when the logic has more than a few inputs. Fortunately, computer-based logic synthesis tools can minimize a large set of BwIean equations efficiently and quickly. Nevertheless, it is important that a designer understand the underlying mathematical description and solution of the problem. This chapter serves as a foundation for your understanding of that important topic and will enable you to execute a manual design of simple circuits, preparing you for skilled use of modern design tools.

Discrete Mathematics for Computing pp Cite as.

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