# Sets In Mathematics

In Mathematics, a set is defined as;
“A collection of well-defined and distinct objects is called set”.
The entries of a set are known as its objects. If you want to get a clear idea about sets, it is necessary that you should have enough idea about well-defined and distinct objects. In Mathematics, well-defined and distinct objects mean those objects which are not equal to other objects. For example, if we say that set of natural numbers, the entries of the natural numbers’ set are 1,2,3,4 and so on. 1, 2, 3 and 4 are well-defined and distinct objects because 1 is not equal to 2 and vice versa. On the other hand, if we say that the set of naughty students in the class, we can’t distinguish between naughty students in the class. In other words, the entries of this set can be equal to each other. As these entries of the set are not well-defined and distinct, therefore, it is not a set.

# How To Represent Sets in Mathematics?

There is also a specific method to represent sets in mathematics. In order to represent sets in mathematics, we have to use ‘Capital alphabets’ i.e A, B, C, D and so on. On the other hand, the entries of a set are written in ‘Small alphabets’ i.e a, b, c, d and so on. Anyhow, the entries of the sets are also written in the form of numerals and symbols etc. These entries of a set are enclosed in the braces or the curly brackets. Some essential examples of the sets are given below;
A = {1, 2, 3, 4}
N = {1, 2, 3, 4, 5…….}
W = {0, 1, 2, 3, 4, 5, 6…….}

# Ways To Represent Sets in Mathematics

There are three possible ways to represent sets in mathematics. These three ways are explained below;

# Tabular Form

The method of representing the set in which we list all the elements of the set and these elements of sets are separated by commas. After listing these elements, we enclose these elements in the curly brackets. Some essential examples of sets that are represented in the tabular form are given below;
A = {6, 7, 8, 9, 10}
E = {2, 4, 6, 8, 10……}
O = {1, 3, 5, 7, 9……}

# Descriptive Form

The process of representing the sets in the form of words is known as descriptive form. Some essential examples of the descriptive form of sets are given below;
N = Set of natural numbers
W = Set of whole numbers
Z = Set of integers
O = Set of odd numbers
E = Set of even numbers

# Set Builder form

The presentation of the sets in the symbolic form is known as set builder form. Some essential examples of the representation of the sets in set builder form are given below;
A = {x | x N ^ x > 10}
B = {x | x W ^ x > 5}
C = {x | x N ^ x > 7}

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# Types of Sets in Mathematics

There are different types of sets in mathematics. These types of sets along with their examples are explained below;

# Empty or Null Set

A set which has no element is known as an empty or null set. In order to represent an empty or null set, we use the symbol {} or . We can represent a null or an empty set in the following way;
A =
B = {}

# Finite Set

A set which has the finite number of elements or the limited number of elements is known as a finite set. Some essential examples of finite sets are given below;
A = Set of first ten natural numbers
B = {1, 2, 3, 4, 5}
C = {x | x W ^ x > 5}

# Infinite Set

A set which has the infinite number of elements or the unlimited number of elements is known as an infinite set. Some essential examples of infinite sets are given below;
N = Set of natural numbers
W = {0, 1, 2, 3, 4, 5……..}

# Equivalent Sets

Two sets are called equivalent sets if these sets have the equal number of elements. We can represent two equivalent sets by using the symbol ↔. Some essential examples of equivalent sets are given below;
A = {1, 2, 3}
B = {a, b, c, d}
C = {2, 4, 5, 6}
D = {x, y, z}
A and D are equivalent sets because these sets have an equal number of elements and we can represent these two sets as A ↔ D. On the other hand, B and C are also equivalent sets. Its reason is that these two sets have an equal number of elements and we can represent these two sets as B ↔ C.

# Equal Sets

Two sets are called equal sets if these two sets have equal elements and these elements are also same. In order to represent two equal sets, we use the symbol ‘=’. Some essential examples of equal sets are given below;
A = {1, 2, 3, 4, 5}
B = {a, b, c, d}
C = {2, 3, 4, 5, 1}
D = {a, b, c, d}
A and C are equal sets because these sets have an equal number of elements and these elements are also the same. We can represent these two sets as A = C. On the other hand, B and D are also equal sets because these two sets have an equal number of elements and these elements are also the same. We can also represent these two sets as B = D.

# Singleton Set

A set which has only one element is known as singleton set. Some essential examples of singleton sets are given below;
A = {11}
B = {a}
C = {0}

# Subset

If A and B are two sets and all the elements of set A are present in the set B, set A is called the subset of set B. Example of the subset is given below;
A = {1, 2, 3}
B = {1, 2, 3, 4}
As all the elements of A are present in the B, therefore, A is called the subset of B.

# Proper subset

If A and B are two sets and A is a subset of B and A ≠ B, A is also called a subset of B. An example of the subset is given below;
A = {1, 2}
B = {1, 2, 3, 4}
Now, we can say that A is the subset of B.

# Improper Subset

If A and B are two sets and A is an improper subset of B only either A = B or A contains at least one such element which is not present in set B. Example of an improper subset is given below;
A = {1, 2, 3, 4}
B = {1, 2, 3, 4}
A is an improper subset of B.
A = {1, 2, 3, 4, 7}
B = {1, 2, 3, 4, 5}
A is improper subset of B.

# Power set

A set which contains all the subsets of a set is known as its power set. The power set of A = {1, 2} is given below;
P (A) = { , {1}, {2}, {1,2} }

# Universal Set

A set which is superset of all the sets under consideration is known as a universal set. An example of universal set is given below;
U = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}
A = {1, 2, 3, 4, 5}
B = {2, 4, 6, 8}
Now, U is called the universal set.

## What is cloning?

In biomedical research, we have to come across with cloning. In biomedical research, the definition of cloning is that the process in which we make duplicates of different kinds of biological materials for the research process is known as cloning. With the help of cloning, it is possible for us to prepare genetically identical copies of a cell or an organism. In other words, we can also define cloning as the process in which a cell replicates itself without any kind of genetic alternation is known as cloning. All the prokaryotic organisms like bacteria have the ability to create its identical copies by budding or binary fission. On the other hand, we are creating clones of all the eukaryotic organisms.

# History of cloning

The concept of cloning is not a new concept in the biomedical science. Its reason is that first of all, a German Scientist Hans Adolf had given the concept of cloning in 1885. According to Genetic Science Learning Center, in 1902, he was able to divide an embryo into two viable embryos. After that, in 1958, a British biologist was able to create clones of a frog from the skin cells of an adult frog. The first clone of a mammal was created in 1996. In 1996, a female sheep gave birth to a Finn Desert Lamb. After preparing first clone of mammal, the scientists were able to find out the ways to prepare more clones of other animals. Later on, scientists were also able to create clones of different cells of human beings. In 2013, the scientists were able to create first clone of human cells and it was clone of stem cell. Nowadays, the scientists are using these stem cells for multiple purposes i.e hair re-growth and treatments from the burns.

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What is artificial intelligence?

# Types of cloning

Most of the people think that there is only one type of cloning and that is organism cloning. They should come to know that there are three types of cloning and these three different types of cloning are explained below;

## Molecular cloning

Molecular cloning is also known as gene cloning. With the help of molecular cloning, it is possible for us to prepare identical copies of DNA molecules in the chromosomes.

## Organism cloning

Organism cloning is also known as reproductive cloning. With the help of organism cloning, it is possible for us to produce an identical copy of an organism.

## Therapeutic cloning

In this essential type of cloning, it is possible for us to create clones of human embryos. These clones are created by producing the stem cells. After creating these clones, it is possible for us to use these clones for the treatment of various diseases.

# Process of cloning

As we know that nowadays, it is possible for us to prepare clones of different kinds of animals but according to Bill Muir (a famous geneticist at the Purdue University), it is easy for us to prepare clones of cow than other animals. Its reason is that it is easy for us to obtain eggs from the cows than other animals. Here, we will discuss cloning process in cows. For this reason, first of all, we have to take DNA from the cell of an adult cow. This adult cow is known as donor.
After taking this DNA cell from an adult cow, we insert it in the egg cell of another cow. Before inserting DNA cell in the egg cell, we have to remove nucleus from the egg cell. After removing the nucleus, the egg cell will be able to read and duplicate the DNA of the donor cell. It is necessary for us to zap the newly born embryo cell with the help of electricity and it starts the multiplying process. When this newly-created embryo becomes a blastocyst, we implant it in the surrogate mother. As a result, a newly-born cow will be identical to the donor cow.

While studying cloning, there come some questions into our mind. If we want to learn it completely, it is necessary for us to get answers to these questions. Here, we will discuss some questions along with their answers about cloning.

# Do clones ever occur naturally?

The answer to this question is “Yes”. There are some single celled organisms like bacteria that can reproduce identical copies of their cells with the help of asexual reproduction.

# What sort of cloning research is going on at NHGRI?

The most common type of cloning research that is done by the researchers at the NHGRI is gene cloning.

# Do cloned animals always look identical?

No, it is not possible that the cloned animals always look identical. Anyhow, these clones always share the same genetic material.

## Top 10 Search Engines In The World

A web search engine is also known as an internet search engine. It is designed in such a way that it provides us with a platform to search a query or worldwide web in a systematic way. A search engine shows these search results in a line of results. These results are known as SERPs (Search Engine Results Pages). These search results pages show the information in the form of images, articles, videos and research papers etc. A search engine uses algorithms or web crawlers to show information. When we are going to find any kind of information on the internet, we have to use these search engines. There are different mathematical formulas of all the search engines for showing results. There are unique algorithms of all the search engines. It means that if a web page has top ranking in Google, it is not necessary that it should also have a top ranking in the Bing or Yahoo search engine. Another essential fact about these search engines is that they keep on changing in their algorithms constantly. There are lots of search engines are available that are helpful for us to search for data on the internet. Here, we will provide a list of top 10 search engines in the world.

# Top 10 Search Engines In The World

1)    According to HubSpot and Internet Live Stats, the estimated search queries per second are 70,000 on Google. It means that there are almost 5.8 billion search queries on Google and annually, there are 2 trillion search queries on Google.
2)    According to Google search engine stats, 15% searches on Google are those searches that are not seen ever before on the Google.
3)    According to Backlinko, Google checks almost 200 different factors of a website before showing it in the best results for the users.
4)    According to the report of Statista, almost 63% search queries on Google are made through mobiles.
5)    It is estimated that Google’s ad revenue in 2018 was almost \$116 billion.
6)    In the second quarter of 2019, almost 93% of search queries in the US are made through Google search engine.

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# Bing

Bing is an essential search engine and it is a product of Bing. Microsoft is working hard on this search engine but they are still not able to attract the users that Bing is also a reliable search engine just like Google. As it is the product of Microsoft, therefore, it is the default search engine in the Windows PCs. Despite this support of Microsoft, Bing is not able to increase its market share from 6%. The market share of Bing search engine is in between 2.34% to 5.29%. If you don’t like Google search engine and if you are looking for an alternative of the Google search engine, Bing is the best choice for you.

# Yahoo

Most of the internet users are well aware of Yahoo because it is one of the best email providers for users. On the basis of the market share, it is also holding third place in the list of top search engines. The average market share of this search engine is in between 1.64% to 2.04%. Yahoo search engine is gaining lots of support from Mozilla Firefox. Its reason is that it is the default program in the Mozilla Firefox. The web portal of Yahoo is very popular among the people. That’s why according to Alexa, it is at the 11th position regarding the most visited websites in all around the world.

# Baidu

Baidu is also an essential search engine and it is the product of China. It was founded in 2000 and due to its familiarity among the users, it has become the fourth most popular search engine. The most interesting fact about Baidu is that it has become the most popular search engine in China. According to Wikipedia, Baidu is serving billions of search queries to the people. As a content creator, if you want to rank your post on the Baidu, you will have to spend more efforts on SEO than Google. The global market share of this search engine is just 0.92%. Anyhow, the most important fact about this search engine is that almost 90% of Chinese are using Baidu for their search queries.

# Yandex.ru

According to Alexa, Yandex is one of the most popular websites on the internet. On the basis of search results for this website, this website is positioned at the 30th position in Alexa ranking. On the other hand, it is also known as the most popular website in Russia and in Russia, its Alexa rank is 4. The most interesting fact about Yandex is that it is the most popular search engine in Russia. Its reason is that its market share in Russia is almost 65%. Anyhow, if we take an overview of its global market share, we come to know that it has very little market share. It means that the market share of Yandex is just 0.47% to 0.83%.

# DuckDuckGo

The average searches of a day on this search engine are 47 million. In spite of these 47 million searches, the average market share of this essential software is just below 0.5%. This essential search engine doesn’t have its own algorithms like Google and Bing. Therefore, in order to show results against any query, it has to depend on other search engines. Anyhow, the most interesting thing about this search engine is that it has a very clean interface and it is not loaded with enough ads like Google.

Ask.com is basically a question answers website. On this website, users can post their questions and interesting people can provide the best answers to these questions. Anyhow, this essential website has also the functionality of a search engine. That’s why it has also a 0.42% market share of the search queries.

# AOL.com

In the past, this search engine was very famous among the users. Anyhow, during recent years, it has lost its reputation but it is still able to manage its place in the top 10 most famous search engines. Its reason is that the net market share of traffic of this search engine is 0.05%.

# WolframAlpha

The functionality of this search engine is different from other search engines. Its reason is that this search engine is famous among the researchers as a computational search engine. It means that with the help of this essential search engine, it is possible for the users to get information about different facts and numbers about different topics. The users are also using it in order to do lots of calculations. This search engine uses lots of assumptions in order to make these kinds of calculations for the users.

# Internet Archive

The functionality of this search engine is also different from other kinds of search engines. Its reason is that with the help of this search engine, it is possible for the users to get an idea about the complete history of a website. This search engine has the ability to provide a complete history of a website like its domain and website look from 1996 to now.

## What Is Artificial Intelligence?

AI (Artificial Intelligence) is a wide ranging branch of computer science and it is concerned with the building of such smart machines that are capable to perform tasks just like human beings. Artificial intelligence is progressing very rapidly from the SIRI to self-driving cars. Artificial intelligence is divided into two categories. First is known as weak AI. Weak AI means that scientists are working on the preparation of such machines which are able to perform narrow tasks like a facial reorganization. The second type is known as strong AI. Strong AI means that scientists are working on the preparation of such machines that are able to perform some specific tasks like playing chess or solving equations.

# Main components and characteristics of artificial intelligence

In order to operate different kinds of intelligent systems, it is necessary for us to get an idea about the main components of artificial intelligence. Main components or characteristics of artificial intelligence are explained below;

# 1.     Feature engineering

The process in which we identify the minimal number of informative features from a dataset is known as feature extraction. It is possible for us to improve the performance of the machine learning with the help of these informative features. For this reason, first of all, we should reduce the entropy of the system because it is helpful for us to increase the information gain. Secondly, we should try to ensure zero correlation among these informative features of the machines.

# 2.     Artificial neural networks

There is a set of computing nodes in the consecutive layers of the artificial machines. The main function of these neural networks is to provide weighted interconnections between these nodes and consecutive layers of the artificial machines. With the help of these neural networks, it is possible for us to reduce the optimal weight of the connections. Its reason is that with the help of these artificial neural networks, these connections can adopt common-weight sharing strategy. Some essential kinds of artificial neural networks are CNN (Convolution Neural Networks) and RNN (Recurrent Neural Networks) etc.

# 3.     Deep learning

Between input and output layers of artificial machines, there are layers of deep learning. With the help of these deep learning frameworks, it is possible for the artificial machines to perform automatic feature extractions along with classification learning. With the help of these deep layers, it is also possible for us to decrease the learning time of the artificial machines.

# Applications of artificial intelligence

Nowadays, we are using artificial intelligence in almost all fields of life. Some essential applications of artificial intelligence are given below;

# 1.     SIRI

Almost all the people are familiar with the personal assistant of Apple. Her name is SIRI. We have to interact with this friendly voice-activated computer on a daily basis. SIRI is helping us in various ways like with the help of her assistance, it is possible for us to find the required amount of information, to add some specific events in the calendars, to get directions from her and so on. In order to understand our questions and to provide possible answers to these questions, SIRI is using machine learning technology.

# 2.     Alexa

Alexa is helpful for those people who have limited mobility. Its reason is that with the help of machine learning abilities of Alexa, it is possible for us to scour the web for information, to shop anything, to schedule the appointments, to set alarms and other millions of tasks.

# 3.     Tesla

If you have not used Tesla yet, it means that you have missed a lot. Its reason is that Tesla is known as the best driving car that is ever made. Its reason is that this essential kind of driving car has lots of features like it has lots of predictive capabilities, it has self-driving features and it is also using sheer technology that is known as coolness. If you are interested in the field of technology and you can easily own cars, it is necessary for you to own Tesla.

# 4.     Cogito

Nowadays, we are receiving millions of phone calls that are based on artificial intelligence. For this reason, companies are using Cogito technology. The most important benefit of using this technology for making phone calls is that it is possible for us to improve the emotional intelligence of the customers. As a result, these companies are able to improve customer relationship.

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# 5.     Netflix

If you are interested in watching movies, you will be familiar with Netflix. Netflix is using highly accurate predictive technology. By using this technology, it is possible for Netflix to analyze the reactions of the customers. After analyzing their reactions, it shows the best results for their favourite movies. As the dataset of this essential technology is growing over the years, therefore, it is getting smarter and smarter. The only drawback of this technology is that it is promoting only big-named movies and small-labelled movies remain unnoticed.

# 6.     Nest

Almost all the people are familiar with the Nest. Its reason is that we are using Nest in our homes and offices for controlling temperatures according to our needs. Now, the most important quality of this artificial technology is that it is easy for us to control it with the help of voice.

There are lots of benefits of artificial intelligence. Some essential advantages of artificial intelligence are explained below;

# 1.     Reduction in human error

As a human being, there are chances of making some mistakes. On the other hand, if we are operating anything with the help of artificial intelligence, there will be no chance of error. Its reason is that artificial intelligence is performing different tasks on the basis of previously gathered information. You just need to operate the artificial machines by proper programming.

# 2.     Takes risks instead of humans

It is a fact that there are some tasks that are very risky for human beings. For example, it is risky for human beings to go on Mars, to defuse a bomb and to explore the deepest parts of the ocean. For this reason, we can prepare robots and these robots can do these tasks for us.

# 3.     Available 24X7

It is also a fact that it is impossible for a human being to work actively 24 hours of the day. Its reason is that a human being requires breaks for the preparation of the next day’s work. Under such a situation, a human being can work 6 to 8 hours in a day. On the other hand, artificial machines can work around the clock without any break. Therefore, we can say that these artificial machines are available 24 hours of the day.

# 4.     Helpful in repetitive jobs

In our daily life, we have to perform different kinds of repetitive tasks and sometimes, these repetitive tasks are boring for us. With the help of artificial machines, it is possible for us to perform these repetitive tasks. While performing these repetitive tasks, these machines will never make any kind of mistake. On the other hand, while performing repetitive tasks, a human being can make some mistakes.

# 5.     Digital assistance

There are lots of companies that are using digital assistants in order to interact with their customers. With the help of this digital assistance, it is also possible for these companies to decrease the needs of human assistance. Moreover, this digital assistance is available for the assistance of the customers around the clock without any time limitation.

Along with the advantages of artificial intelligence, there are also some disadvantages of artificial intelligence. These disadvantages are explained below;

# 1.     High costs of creation

As we know that these artificial machines are very complex and in order to create these artificial machines, there require huge costs. Moreover, we are also observing lots of updates in the field of artificial intelligence. Therefore, if we want to meet the latest requirements of AI, it is also necessary for us to update hardware and software.

# 2.     Making humans lazy

As we know that with the help of artificial intelligence, it is possible for us to automate the majority of the work. As a result, human beings are becoming addicts of these machines and this thing can create lots of problems for future generations.

# 3.     Unemployment

It is also a fact that artificial intelligence is also taking positions of human beings in most of the companies. Therefore, we can also say that the developments in the field of artificial intelligence can also create lots of unemployment issues for human beings.

# 4.     No emotions

Nowadays, we are using artificial machines at work places. While working at a workplace, it is necessary for us to work by making a team. These machines can’t develop bonds with human beings. As a result, it is almost impossible for us to work by making a team with these artificial machines.

# 5.     Lacking out of box thinking

One of the major drawbacks of these artificial machines is that these artificial machines can only perform those tasks that are programmed in them. It means that it is almost impossible for us to increase productivity within a company by using artificial machines.

## Difference Between Software And Hardware

As we know that this is an era of computer and technology. Therefore, all computer users have to come across terms of hardware and software. When we think about hardware and software, different questions come to our minds like ‘What is the relation between software and hardware?’ and ‘What is the difference between software and hardware?’ etc. Hardware and software are two major components of a computer. Here, we will discuss the difference between hardware and software.

# What is the hardware?

The physical part of a computer system that we can touch is known as hardware. All the computer systems require hardware for their existence. Some essential examples of computer hardware are CPU, Mouse and Keyboard etc.

# What is software?

A collection of instructions, procedures and documents that are required to run different functions of a computer is known as software. Like computer hardware, we can’t touch computer software. Some essential examples of computer software are MS Word, PowerPoint, Google Chrome and Excel etc.

# Difference between software and hardware

After getting an idea about the basic definitions of computer hardware and software, we are in a better position to understand the differences between software and hardware of the computer. Their differences along with the basis of differentiation are explained below;

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# Difference between definitions of software and hardware

### Hardware

Hardware refers to all the physical and tangible parts of a computer. These parts of the computer are helpful for us to store information and to execute the software.

### Software

A collection of programs, documents and codes that are required to interact with the computer systems is known as software. Generally, it is a program that is required for a computer system to perform different kinds of tasks.

# Difference between the development of software and hardware

### Hardware

It is possible for us to develop computer hardware with the help of electronic components and other materials.

### Software

We can develop computer software with the help of a computer programming language. For this reason, we have to develop computer software by writing instructions in the programming language.

# Difference between types of software and hardware

### Hardware

There are different kinds of computer hardware like input devices (devices that are used to enter data into computer system like Mouse, Keyboard and Scanner etc), output devices (devices that are used to show output on the computer system like monitor, speaker and printer etc), control devices, processing unit and storage devices etc.

### Software

Like hardware, there are also different kinds of computer software like programming software (that is helpful for the developers to develop other software and computer applications), system software (system software is required to run computer hardware and applications) and application software (it is a program or a set of programs that are designed for the end-users) etc.

# Difference between the functionality of software and hardware

### Hardware

The main function of computer hardware is to provide a platform for the operation of computer software.

### Software

The main function of computer software is to enable the computer user to perform different tasks with the help of computer software.

# Replacement

While using computer software and hardware, there is a possibility that it may be damaged. Under such a situation, we can repair computer software and hardware in the following ways;

### Hardware

If computer hardware is damaged, it is necessary for us to replace it with a new one.

### Software

We can easily replace computer software just by reinstalling it.

# Difference between examples of software and hardware

### Hardware

The main examples of computer hardware are CPU, Monitor, Keyboard, Mouse, Speaker and Printer etc.

### Software

The main examples of computer software are Google Chrome, Mozilla Firefox, Microsoft Windows, MS Office and Google Apps etc.

# Manufacturing and development

### Hardware

Computer hardware is manufactured.

### Software

Computer software is developed and engineered by software engineers.

# Difference between the nature of software and hardware

### Hardware

The nature of computer hardware is tangible and physical.

### Software

The nature of computer software is non-tangible and logical.

# Difference between inter-dependency of software and hardware

Software and hardware depend upon each other in the following ways;

### Hardware

If we want to start the functionality of computer hardware, it is necessary for us to load computer software correctly on it.

### Software

If we want to operate specific computer software, it is also necessary for us to install that software on specific hardware.

# Difference between wear and tear of software and hardware

### Hardware

Over time, the performance of computer hardware depreciates.

### Software

There will be no chance in depreciation of the computer software over time. Anyhow, if you are using specific computer software constantly for a long period of time, there is a possibility that there may appear bugs in this software.

# Virus attacks

There is also the different impact of the virus attacks on hardware and software.

### Hardware

There is no direct impact of the virus attacks on computer hardware.

### Software

These virus attacks can last direct impacts on computer software programs and applications.

# Difference between the transformation of hardware and software

### Hardware

It is not possible for us to transfer computer hardware from one place to another place in the form of electrical signals.

### Software

It is possible for us to transfer computer software from one place to another place in the form of electrical signals.
After understanding the main difference between hardware and software, it is also necessary for you to get answers to the following questions.

# How do hardware and software work together?

Computer hardware and computer software depend upon each other. Anyhow, it is possible for us to run computer hardware without computer software but we can’t run computer software without computer hardware. For example, if you have a video game in the form of computer software, you can’t operate it without computer hardware. Its reason is that for the operation of the computer software there requires memory, processors, video cards and sound cards etc.