Computer Applications

computers are used in various fields, such as homes, businesses, government offices, research organizations, educational institutions, medical, entertainment, etc. Computers have taken industries and businesses to a whole new level. In this article, we have elaborated the most common uses of computers in different fields:

Business

Currently, computers can be seen in almost every business. Computers are almost part of a business setup because they increase productivity and help race in a competitive environment. In businesses, computers are primarily used to store and manage accounts and personal data, maintain projects, track inventory status, and make reports and presentations. Besides, computers are best suited for transaction processing because they are more accurate and faster than humans. Computers also help people analyze their investment, expenses, profits, sales and many other aspects of the business.

Science

Scientists are amongst one of those people who use computers as their primary work tool. In science, research and engineering, computers are best suited for collecting, analyzing, categorizing, and storing the data. They also help scientists to exchange data with each other both internally and internationally. Computers enable scientists from different locations (such as different countries) to work together on the same project with cloud support. Besides, computers play a crucial role in launching, maintaining, controlling spacecraft, and operating many other technologies.

Government

In the government sector, computers are beneficial. They are getting used to performing various functions in different departments and improving their services' quality, efficiency, and productivity. Some examples of such services are city planning, traffic control, law enforcement, infrastructure developments, and tourism. In most cases, the primary purposes of using computers are performing data processing tasks, maintaining citizens' database, and promoting a paperless environment. Apart from this, computers are playing a crucial role in the country's defense system. They are helping in missile development, rocket, satellite launches, etc.

Health and Medical

Computers are radically changing the methods of diagnosis in hospitals. They are used for maintaining patients' information, records, live monitoring of patients, X-rays, and more. Everything is being digitized with the help of computers. Computers help configure lab-tools, monitor heart rate, and blood pressure, etc. Doctors get extra advantages in treating patients with proper drugs and medicines. Additionally, computers enable doctors to exchange patient's data easily with other medical specialists. Besides, advanced surgical devices are based on robotics that helping surgeons to conduct complex operations and surgeries remotely.

Education

Computers are broadly getting used in the education field. They help people get different educational materials (such as images, videos, e-books, etc.) in one place. All such information can be accessed through the Internet. Additionally, computers are best suited for online classes, online tutoring, online examinations, and creating assignments and projects. Apart from this, they can also be used to maintain and monitor student performance and other information.

Industry

Computers are used in industries to perform various tasks, such as maintaining inventory, interior designing, designing samples or virtual products, communicating over video conferencing, and more. Online marketing has made it easier for people to buy products in rural areas. Online trading in stock markets has also seen a significant revolution due to its easy participation potential. Computers have enabled people from different levels of different locations to participate easily in stock marketing.

Banking

Banking has become so advanced in the past few years. Most countries use online banking systems where customers can access their data directly using computers and the Internet. People can check their account balance, transfer money, and pay online bills, including credit cards. Besides, Banks use computers to perform transactions and store customer data, transaction records, etc. Banks have reduced the number of manual errors, number of employees, and costs to a great extent by using computers. ATMs are the best example of computers that are helping people to withdraw and deposit the money themselves.

Entertainment

Computers nowadays are one of the best mediums for entertainment. Computers can be used to watch movies, play games, listen to music, etc. Computers combined with MIDI instruments can be used to record audio through artificial instruments. Besides, people can also enjoy recording their videos with webcam and apply several entertaining AI effects. Several Photo editor programs are also available with fabulous powerful features.

Training

Most companies use computers to provide training to their employees. Computer-based training helps companies save their time, money, and increase productivity. Also, computer-based training can be used to train employees for large distances in various locations. This will eliminate travel time and costs, making the training process much more comfortable and smoother.

Arts

Computers have become part of art, photography, dance, and culture. Computers with advanced features allow users to draw their projects directly on it. Besides, people can use computers to digitize their photos. There are several photo editor software that can help people edit and customize their photos. Apart from this, the dance's movements and steps can be shown live with animations' help.

Sports

In today's technologically developed world, computers are being used in almost every sport. There are many sports activities where computers are making things possible. In sports, computers are mainly used to maintain scoreboards, records, and other statistics. Furthermore, they are used to analyze player movements and make various in-game decisions. Computers help make complex in-game decisions (especially in umpiring), which cannot be seen by human eyes.

Robotics

Robotics is one of the emerging fields of technology that uses computers for science and engineering as well as designing machines. These machines can be virtual (such as software bots) and physical ones that can reduce or eliminate human workload. Additionally, some machines can perform heavy tasks that humans cannot complete, or that may take a long time to complete. Car manufacturing was one of the first examples where robots helped to assemble car parts and perform many other heavy tasks. However, nowadays, robots are beneficial in many fields, such as exploring areas where conditions are difficult for humans, helping the military, helping law enforcement and helping health professionals, etc.

Uses of Computer

Computers are playing a vital role in almost every field and making our day-to-day tasks more manageable. Computers were only used to perform complex numerical calculations in a previous time, but they have reached too far and now perform many different roles. They are now performing diverse set functions from complicated calculations to generating business reports, bill generation to education, programming or development to entertainment, etc.

 

Computer Overview

Today’s world is an information-rich world and it has become a necessity for everyone to know about computers. A computer is an electronic data processing device, which accepts and stores data input, processes the data input, and generates the output in a required format.

A computer system has three main components: hardware, software, and people. The equipment associated with a computer system is called hardware. Software is a set of instructions that tells the hardware what to do. People, however, are the most important component of a computer system - people use the power of the computer for some purpose. In fact, this course will show you that the computer can be a tool for just about anyone from a business person, to an artist, to a housekeeper, to a student - an incredibly powerful and flexible tool.

The purpose of this tutorial is to introduce you to Computers and its fundamentals.

Functionalities of a Computer

If we look at it in a very broad sense, any digital computer carries out the following five functions −

Step 1 − Takes data as input.

Step 2 − Stores the data/instructions in its memory and uses them as required.

Step 3 − Processes the data and converts it into useful information.

Step 4 − Generates the output.

Step 5 − Controls all the above four steps.

Characteristics of Computer

• Speed 
• Storage
• Accuracy 
• Diligence
• Cost Effectiveness 
• Flexibility

Advantages of computer

Multitasking

Multitasking is one of the major advantage of computer. Person can perform multiple task, multiple operation, calculate numerical problems within few seconds. Computer can perform trillion of instructions per second.

Speed

Now computer is not just a calculating device. Now a day’s computer has very important role in human life. One of the main advantages of computer is its incredible speed, which helps human to complete their task in few seconds. All the operations can be performed very fast just because of its speed elsewise it takes a long time to perform the task.

Cost/ Stores huge amount of data

It is a low cost solution. Person can save huge data within a low budget. Centralized database of storing information is the major advantage that can reduce cost.

Accuracy

One of the root advantage of computer is that can perform not only calculations but also with accuracy.

Data Security

Protecting digital data is known as data security. Computer provide security from destructive forces and from unwanted action from unauthorized users like cyber attack or access attack.

 

Disadvantage of Computer

As we know advantage comes with disadvantage.

Virus and hacking attacks

Virus is a worm and hacking is simply an unauthorized access over computer for some illicit purpose. Virus is being transferred from email attachment, viewing an infected website advertisement, through removable device like USB etc. once virus is transferred in host computer it can infect file, overwrite the file etc.
For example: Huge portion of internet  was going down including Twitter, Netflix, Reddit and CNN in October 2016 because the largest DDoS attack was launched on service provider DYN using IoT Botnet.

Online Cyber Crimes

Online cyber-crime means computer and network may have used in order to commit crime. Cyberstalking and Identity theft are the points which comes under online cyber-crimes. For example: one may get the access of the access to your shopping account like amazon account now that person will be able to know your personal details like debit card or credit card number which can be than misused.

Reduction in employment opportunity

Mainly past generation was not used of the computer or they have the knowledge of computer they faced a big problem when computer came in field. As we have seen in banking sector senior bank employees faced this problem when computer came to the banking sector.
Above were the main disadvantage of computer, no IQ, Dependency, No feeling, Break down are the basic disadvantages of computer.

 

 

Logic Gates

A logic gate is a device that acts as a building block for digital circuits. They perform basic logical functions that are fundamental to digital circuits. Most electronic devices we use today will have some form of logic gates in them. For example, logic gates can be used in technologies such as smartphones, tablets or within memory devices.

In a circuit, logic gates will make decisions based on a combination of digital signals coming from its inputs. Most logic gates have two inputs and one output. Logic gates are based on Boolean algebra. At any given moment, every terminal is in one of the two binary conditions, false or true. False represents 0, and true represents 1. Depending on the type of logic gate being used and the combination of inputs, the binary output will differ. A logic gate can be thought of like a light switch, wherein one position the output is off -- 0, and in another, it is on -- 1. Logic gates are commonly used in integrated circuits (IC).

Basic logic gates

There are seven basic logic gates: AND, OR, XOR, NOT, NAND, NOR, and XNOR.

The AND gate is so named because, if 0 is called "false" and 1 is called "true," the gate acts in the same way as the logical "and" operator. The following illustration and table show the circuit symbol and logic combinations for an AND gate. (In the symbol, the input terminals are at left and the output terminal is at right.) The output is "true" when both inputs are "true." Otherwise, the output is "false." In other words, the output is 1 only when both inputs one AND two are 1.

 

AND gate

Input 1	Input 2	Output
	1
1
1	1	1

The OR gate gets its name from the fact that it behaves after the fashion of the logical inclusive "or." The output is "true" if either or both of the inputs are "true." If both inputs are "false," then the output is "false." In other words, for the output to be 1, at least input one OR two must be 1.

 

OR gate

Input 1	Input 2	Output
	1	1
1		1
1	1	1

 

The XOR ( exclusive-OR ) gate acts in the same way as the logical "either/or." The output is "true" if either, but not both, of the inputs are "true." The output is "false" if both inputs are "false" or if both inputs are "true." Another way of looking at this circuit is to observe that the output is 1 if the inputs are different, but 0 if the inputs are the same. 

 

 

XOR gate

Input 1	Input 2	Output
	1	1
1		1
1	1

 

A logical inverter, sometimes called a NOT gate to differentiate it from other types of electronic inverter devices, has only one input. It reverses the logic state. If the input is 1, then the output is 0. If the input is 0, then the output is 1.  

 

 

 

Inverter or NOT gate

Input	Output
1
	1

 

The NAND gate operates as an AND gate followed by a NOT gate. It acts in the manner of the logical operation "and" followed by negation. The output is "false" if both inputs are "true." Otherwise, the output is "true."

 

NAND gate

Input 1	Input 2	Output
		1
	1	1
1		1
1	1

 

The NOR gate is a combination OR gate followed by an inverter. Its output is "true" if both inputs are "false." Otherwise, the output is "false."

NOR gate

Input 1	Input 2	Output
		1
	1
1
1	1

 

The XNOR (exclusive-NOR) gate is a combination XOR gate followed by an inverter. Its output is "true" if the inputs are the same, and "false" if the inputs are different.

 

XNOR gate

Input 1	Input 2	Output
		1
	1
1
1	1	1

Complex operations can be performed using combinations of these logic gates. In theory, there is no limit to the number of gates that can be arrayed together in a single device. But in practice, there is a limit to the number of gates that can be packed into a given physical space. Arrays of logic gates are found in digital ICs. As IC technology advances, the required physical volume for each individual logic gate decreases and digital devices of the same or smaller size become capable of performing ever-more-complicated operations at ever-increasing speeds.

Composition of logic gates

High or low binary conditions are represented by different voltage levels. The logic state of a terminal can, and generally does, often change as the circuit processes data. In most logic gates, the low state is approximately zero volts (0 V), while the high state is approximately five volts positive (+5 V).

Logic gates can be made of resistors and transistors or diodes. A resistor can commonly be used as a pull-up or pull-down resistor. Pull-up and pull-down resistors are used when there are any unused logic gate inputs to connect to a logic level 1 or 0. This prevents any false switching of the gate. Pull-up resistors are connected to Vcc (+5V), and pull-down resistors are connected to ground (0 V).

Commonly used logic gates are TTL and CMOS. TTL, or Transistor-Transistor Logic, ICs will use NPN and PNP type Bipolar Junction Transistors. CMOS, or Complementary Metal-Oxide-Silicon, ICs are constructed from MOSFET or JFET type Field Effect Transistors. TTL IC's may commonly be labeled as the 7400 series of chips, while CMOS ICs may often be marked as a 4000 series of chips.

 

Memory

Memory is the electronic holding place for the instructions and data a computer needs to reach quickly. It's where information is stored for immediate use. Memory is one of the basic functions of a computer, because without it, a computer would not be able to function properly. Memory is also used by a computer's operating system, hardware and software.

There are technically two types of computer memory: primary and secondary. The term memory is used as a synonym for primary memory or as an abbreviation for a specific type of primary memory called random access memory (RAM). This type of memory is located on microchips that are physically close to a computer's microprocessor.

If a computer's central processer (CPU) had to only use a secondary storage device, computers would become much slower. In general, the more memory (primary memory) a computing device has, the less frequently the computer must access instructions and data from slower (secondary) forms of storage.

Memory vs. storage

The concept of memory and strorage can be easily conflated as the same concept; however, there are some distinct and important differences. Put succinctly, memory is primary memory, while storage is secondary memory. Memory refers to the location of short-term data, while storage refers to the location of data stored on a long-term basis.

Memory is most often referred to as the primary storage on a computer, such as RAM. Memory is also where information is processed. It enables users to access data that is stored for a short time. The data is only stored for a short time because primary memory is volatile, meaning it isn't retained when the computer is turned off.

The term storage refers to secondary memory and is where data in a computer is kept. An example of storage is a hard drive or a hard disk drive (HDD). Storage is nonvolatile, meaning the information is still there after the computer is turned off and then back on. A running program may be in a computer's primary memory when in use -- for fast retrieval of information -- but when that program is closed, it resides in secondary memory or storage.

How much space is available in memory and storage differs as well. In general, a computer will have more storage space than memory. For example, a laptop may have 8 GB of RAM while having 250 GB of storage. The difference in space is there because a computer will not need fast access to all the information stored on it at once, so allocating approximately 8 GB of space to run programs will suffice.

The terms memory and storage can be confusing because their usage today is not always consistent. For example, RAM can be referred to as primary storage -- and types of secondary storage can include flash memory. To avoid confusion, it can be easier to talk about memory in terms of whether it is volatile or nonvolatile -- and storage in terms of whether it is primary or secondary.

How does computer memory work?

When a program is open, it is loaded from secondary memory to primary memory. Because there are different types of memory and storage, an example of this could be a program being moved from a solid-state drive (SSD) to RAM. Because primary storage is accessed faster, the opened program will be able to communicate with the computer's processor at quicker speeds. The primary memory can be accessed immediately from temporary memory slots or other storage locations.

Memory is volatile, which means that data in memory is stored temporarily. Once a computing device is turned off, data stored in volatile memory will automatically be deleted. When a file is saved, it will be sent to secondary memory for storage.

There are multiple types of memory available to a computer. It will operate differently depending on the type of primary memory used, but in general, semiconductor-based memory is most associated with memory. Semiconductor memory will be made of integrated circuits with silicon-based metal-oxide-semiconductor (MOS) transistors.

Types of computer memory

In general, memory can be divided into primary and secondary memory; moreover, there are numerous types of memory when discussing just primary memory. Some types of primary memory include the following

• Cache Memory. This temporary storage area, known as a cache read as "cash" is more readily available to the processor than the computer's main memory source. It is also called CPU memory because it is typically integrated directly into the CPU chip or placed on a separate chip with a bus interconnect with the CPU.

• RAM. The term is based on the fact that any storage location can be accessed directly by the processor.

• Dynamic RAM. DRAM is a type of semiconductor memory that is typically used by the data or program code needed by a computer processor to function.

• Static RAM. SRAM retains data bits in its memory for as long as power is supplied to it. Unlike DRAM, which stores bits in cells consisting of a capacitor and a transistor, SRAM does not have to be periodically refreshed.

• Double Data Rate SDRAM. DDR SRAM is SDRAM that can theoretically improve memory clock speed to at least 200 MHz.

• Double Data Rate 4 Synchronous Dynamic RAM. DDR4 RAM is a type of DRAM that has a high-bandwidth interface and is the successor to its previous DDR2 and DDR3 versions. DDR4 RAM allows for lower voltage requirements and higher module density. It is coupled with higher data rate transfer speeds and allows for dual in-line memory modules (DIMMS) up to 64 GB.

• Rambus Dynamic RAM. DRDRAM is a memory subsystem that promised to transfer up to 1.6 billion bytes per second. The subsystem consists of RAM, the RAM controller, the bus that connects RAM to the microprocessor and devices in the computer that use it.

• Read-only memory. ROM is a type of computer storage containing nonvolatile, permanent data that, normally, can only be read and not written to. ROM contains the programming that enables a computer to start up or regenerate each time it is turned on.

• Programmable ROM. PROM is ROM that can be modified once by a user. It enables a user to tailor a microcode program using a special machine called a PROM programmer.

• Erasable PROM. EPROM is programmable read-only memory PROM that can be erased and re-used. Erasure is caused by shining an intense ultraviolet light through a window designed into the memory chip.

• Electrically erasable PROM. EEPROM is a user-modifiable ROM that can be erased and reprogrammed repeatedly through the application of higher than normal electrical voltage. Unlike EPROM chips, EEPROMs do not need to be removed from the computer to be modified. However, an EEPROM chip must be erased and reprogrammed in its entirety, not selectively.

• Virtual Memory. A memory management technique where secondary memory can be used as if it were a part of the main memory. Virtual memory uses hardware and software to enable a computer to compensate for physical memory shortages by temporarily transferring data from RAM to disk storage

Sequential and Combinational Circuits

A Sequential circuit combinational logic circuit that consists of inputs variable (X), logic gates (Computational circuit), and output variable (Z).

Combinational circuit produces an output based on input variable only, but Sequential circuit produces an output based on current input and previous input variables. That means sequential circuits include memory elements which are capable of storing binary information. That binary information defines the state of the sequential circuit at that time. A latch capable of storing one bit of information.

There are two types of input to the combinational logic :

1. External inputs which not controlled by the circuit.
2. Internal inputs which are a function of a previous output states.

Secondary inputs are state variables produced by the storage elements, where as secondary outputs are excitations for the storage elements.

Types of Sequential Circuits – There are two types of sequential circuit :

• Asynchronous sequential circuit – These circuit do not use a clock signal but uses the pulses of the inputs. These circuits are faster than synchronous sequential circuits because there is clock pulse and change their state immediately when there is a change in the input signal. We use asynchronous sequential circuits when speed of operation is important and independent of internal clock pulse.But these circuits are more difficult to design and their output is uncertain.
• Synchronous sequential circuit – These circuit uses clock signal and level inputs (or pulsed) (with restrictions on pulse width and circuit propagation). The output pulse is the same duration as the clock pulse for the clocked sequential circuits. Since they wait for the next clock pulse to arrive to perform the next operation, so these circuits are bit slower compared to asynchronous. Level output changes state at the start of an input pulse and remains in that until the next input or clock pulse.

  We use synchronous sequential circuit in synchronous counters, flip flops, and in the design of MOORE-MEALY state management machines.We use sequential circuits to design Counters, Registers, RAM, MOORE/MEALY Machine and other state retaining machines.

Combinational Circuits

A combinational circuit comprises of logic gates whose outputs at any time are determined directly from the present combination of inputs without any regard to previous inputs.

A combinational circuit performs a specific information-processing operation fully specified logically by a set of Boolean functions.

The basic components of a combinational circuit are: input variables, logic gates, and output variables.

Design procedure of a Combinational Circuit

The design procedure of a combinational circuit involves the following steps:

1. The problem is stated.
2. The total number of available input variables and required output variables is determined.
3. The input and output variables are allocated with letter symbols.
4. The exact truth table that defines the required relationships between inputs and outputs is derived.
5. The simplified Boolean function is obtained from each output.
6. The logic diagram is drawn.

The combinational circuit that performs the addition of two bits is called a half adder and the one that performs the addition of three bits (two significant bits and a previous carry) is a full adder.

 

 

 

Laws of Boolean Algebra

As well as the logic symbols “0” and “1” being used to represent a digital input or output, we can also use them as constants for a permanently “Open” or “Closed” circuit or contact respectively.

A set of rules or Laws of Boolean Algebra expressions have been invented to help reduce the number of logic gates needed to perform a particular logic operation resulting in a list of functions or theorems known commonly as the Laws of Boolean Algebra.

Boolean Algebra is the mathematics we use to analyse digital gates and circuits. We can use these “Laws of Boolean” to both reduce and simplify a complex Boolean expression in an attempt to reduce the number of logic gates required. Boolean Algebra is therefore a system of mathematics based on logic that has its own set of rules or laws which are used to define and reduce Boolean expressions.

The variables used in Boolean Algebra only have one of two possible values, a logic “0” and a logic “1” but an expression can have an infinite number of variables all labelled individually to represent inputs to the expression, For example, variables A, B, C etc, giving us a logical expression of A + B = C, but each variable can ONLY be a 0 or a 1.

Examples of these individual laws of Boolean, rules and theorems for Boolean Algebra are given in the following table.

Truth Tables for the Laws of Boolean

Boolean Expression	Description	Boolean Algebra Law or Rule
A + 1 = 1	A in parallel with closed = “CLOSED”	Annulment
A + 0 = A	A in parallel with open = “A”	Identity
A . 1 = A	A in series with closed = “A”	Identity
A . 0 = 0	A in series with open = “OPEN”	Annulment
A + A = A	A in parallel with A = “A”	Idempotent
A . A = A	A in series with A = “A”	Idempotent
NOT A = A	NOT NOT A (double negative) = “A”	Double Negation
A + A = 1	A in parallel with NOT A = “CLOSED”	Complement
A . A = 0	A in series with NOT A = “OPEN”	Complement
A+B = B+A	A in parallel with B = B in parallel with A	Commutative
A.B = B.A	A in series with B = B in series with A	Commutative
A+B = A.B	invert and replace OR with AND	de Morgan’s Theorem
A.B = A+B	invert and replace AND with OR	de Morgan’s Theorem

The basic Laws of Boolean Algebra that relate to the Commutative Law allowing a change in position for addition and multiplication, the Associative Law allowing the removal of brackets for addition and multiplication, as well as the Distributive Law allowing the factoring of an expression, are the same as in ordinary algebra.

Each of the Boolean Laws above are given with just a single or two variables, but the number of variables defined by a single law is not limited to this as there can be an infinite number of variables as inputs too the expression. These Boolean laws detailed above can be used to prove any given Boolean expression as well as for simplifying complicated digital circuits.

A brief description of the various Laws of Boolean are given below with A representing a variable input.

Description of the Laws of Boolean Algebra

• Annulment Law – A term AND´ed with a “0” equals 0 or OR´ed with a “1” will equal 1
•  
  • A . 0 = 0    A variable AND’ed with 0 is always equal to 0
  • A + 1 = 1    A variable OR’ed with 1 is always equal to 1
•  
• Identity Law – A term OR´ed with a “0” or AND´ed with a “1” will always equal that term
•  
  • A + 0 = A   A variable OR’ed with 0 is always equal to the variable
  • A . 1 = A    A variable AND’ed with 1 is always equal to the variable
•  
• Idempotent Law – An input that is AND´ed or OR´ed with itself is equal to that input
•  
  • A + A = A    A variable OR’ed with itself is always equal to the variable
  • A . A = A    A variable AND’ed with itself is always equal to the variable
•  
• Complement Law – A term AND´ed with its complement equals “0” and a term OR´ed with its complement equals “1”
•  
  • A . A = 0    A variable AND’ed with its complement is always equal to 0
  • A + A = 1    A variable OR’ed with its complement is always equal to 1
•  
• Commutative Law – The order of application of two separate terms is not important
•  
  • A . B = B . A    The order in which two variables are AND’ed makes no difference
  • A + B = B + A    The order in which two variables are OR’ed makes no difference
•  
• Double Negation Law – A term that is inverted twice is equal to the original term
•  
  • A = A     A double complement of a variable is always equal to the variable
•  
• de Morgan´s Theorem – There are two “de Morgan´s” rules or theorems,
•  
• (1) Two separate terms NOR´ed together is the same as the two terms inverted (Complement) and AND´ed for example:  A+B = A . B
•  
• (2) Two separate terms NAND´ed together is the same as the two terms inverted (Complement) and OR´ed for example:  A.B = A + B

 

Other algebraic Laws of Boolean not detailed above include:

• Boolean Postulates – While not Boolean Laws in their own right, these are a set of Mathematical Laws which can be used in the simplification of Boolean Expressions.
•  
  • 0 . 0 = 0    A 0 AND’ed with itself is always equal to 0
  • 1 . 1 = 1    A 1 AND’ed with itself is always equal to 1
  • 1 . 0 = 0    A 1 AND’ed with a 0 is equal to 0
  • 0 + 0 = 0    A 0 OR’ed with itself is always equal to 0
  • 1 + 1 = 1    A 1 OR’ed with itself is always equal to 1
  • 1 + 0 = 1    A 1 OR’ed with a 0 is equal to 1
  • 1 = 0    The Inverse (Complement) of a 1 is always equal to 0
  • 0 = 1    The Inverse (Complement) of a 0 is always equal to 1
•  
• Distributive Law – This law permits the multiplying or factoring out of an expression.
•  
  • A(B + C) = A.B + A.C    (OR Distributive Law)
  • A + (B.C) = (A + B).(A + C)    (AND Distributive Law)
•  
• Absorptive Law – This law enables a reduction in a complicated expression to a simpler one by absorbing like terms.
•  
  • A + (A.B) = (A.1) + (A.B) = A(1 + B) = A  (OR Absorption Law)
  • A(A + B) = (A + 0).(A + B) = A + (0.B) = A  (AND Absorption Law)
•  
• Associative Law – This law allows the removal of brackets from an expression and regrouping of the variables.
•  
  • A + (B + C) = (A + B) + C = A + B + C    (OR Associate Law)
  • A(B.C) = (A.B)C = A . B . C    (AND Associate Law)

Boolean Algebra Functions

Using the information above, simple 2-input AND, OR and NOT Gates can be represented by 16 possible functions as shown in the following table.

Function	Description	Expression
1.	NULL	0
2.	IDENTITY	1
3.	Input A	A
4.	Input B	B
5.	NOT A	A
6.	NOT B	B
7.	A AND B (AND)	A . B
8.	A AND NOT B	A . B
9.	NOT A AND B	A . B
10.	NOT AND (NAND)	A . B
11.	A OR B (OR)	A + B
12.	A OR NOT B	A + B
13.	NOT A OR B	A + B
14.	NOT OR (NOR)	A + B
15.	Exclusive-OR	A . B + A . B
16.	Exclusive-NOR	A . B + A . B

Laws of Boolean Algebra Example No1

Using the above laws, simplify the following expression:  (A + B)(A + C)

Q =	(A + B).(A + C)
	A.A + A.C + A.B + B.C	– Distributive law
	A + A.C + A.B + B.C	– Idempotent AND law (A.A = A)
	A(1 + C) + A.B + B.C	– Distributive law
	A.1 + A.B + B.C	– Identity OR law (1 + C = 1)
	A(1 + B) + B.C	– Distributive law
	A.1 + B.C	– Identity OR law (1 + B = 1)
Q =	A + (B.C)	– Identity AND law (A.1 = A)

 

Then the expression:  (A + B)(A + C) can be simplified to A + (B.C) as in the Distributive law.

Code Conversion

There are many methods or techniques which can be used to convert code from one format to another. We'll demonstrate here the following

• Binary to BCD Conversion
• BCD to Binary Conversion
• BCD to Excess-3
• Excess-3 to BCD

Binary to BCD Conversion

Steps

• Step 1 -- Convert the binary number to decimal.

• Step 2 -- Convert decimal number to BCD.

Example − convert (11101)₂ to BCD.

Step 1 − Convert to Decimal

Binary Number − 11101₂

Calculating Decimal Equivalent −

Step	Binary Number	Decimal Number
Step 1	111012	((1 × 24) + (1 × 23) + (1 × 22) + (0 × 21) + (1 × 20))10
Step 2	111012	(16 + 8 + 4 + 0 + 1)10
Step 3	111012	2910

Binary Number − 11101₂ = Decimal Number − 29₁₀

Step 2 − Convert to BCD

Decimal Number − 29₁₀

Calculating BCD Equivalent. Convert each digit into groups of four binary digits equivalent.

Step	Decimal Number	Conversion
Step 1	2910	00102 10012
Step 2	2910	00101001BCD

Result

(11101)2 =  (00101001)BCD

BCD to Binary Conversion

Steps

• Step 1 -- Convert the BCD number to decimal.

• Step 2 -- Convert decimal to binary.

Example − convert (00101001)_BCD to Binary.

Step 1 - Convert to BCD

BCD Number − (00101001)_BCD

Calculating Decimal Equivalent. Convert each four digit into a group and get decimal equivalent for each group.

Step	BCD Number	Conversion
Step 1	(00101001)BCD	00102 10012
Step 2	(00101001)BCD	210 910
Step 3	(00101001)BCD	2910

BCD Number − (00101001)_BCD = Decimal Number − 29₁₀

Step 2 - Convert to Binary

Used long division method for decimal to binary conversion.

Decimal Number − 29₁₀

Calculating Binary Equivalent −

Step	Operation	Result	Remainder
Step 1	29 / 2	14	1
Step 2	14 / 2	7	0
Step 3	7 / 2	3	1
Step 4	3 / 2	1	1
Step 5	1 / 2	0	1

As mentioned in Steps 2 and 4, the remainders have to be arranged in the reverse order so that the first remainder becomes the least significant digit (LSD) and the last remainder becomes the most significant digit (MSD).

Decimal Number − 29₁₀ = Binary Number − 11101₂

Result

(00101001)BCD = (11101)2

BCD to Excess-3

Steps

• Step 1 -- Convert BCD to decimal.

• Step 2 -- Add (3)₁₀ to this decimal number.

• Step 3 -- Convert into binary to get excess-3 code.

Example − convert (0110)_BCD to Excess-3.

Step 1 − Convert to decimal

(0110)_BCD = 6₁₀

Step 2 − Add 3 to decimal

(6)₁₀ + (3)₁₀ = (9)₁₀

Step 3 − Convert to Excess-3

(9)₁₀ = (1001)₂

Result

(0110)BCD = (1001)XS-3

Excess-3 to BCD Conversion

Steps

• Step 1 -- Subtract (0011)₂ from each 4 bit of excess-3 digit to obtain the corresponding BCD code.

Example − convert (10011010)_XS-3 to BCD.

Given XS-3 number  = 1 0 0 1 1 0 1 0 
Subtract (0011)2   = 1 0 0 1 0 1 1 1
                    --------------------
               BCD = 0 1 1 0   0 1 1 1