MCA 5th sem /MCS-052/Solved Assignment/Principles of Management and Information Systems/2017-2018 New

Q.1.
A.1.

Enterprise resource planning (ERP) is an enterprise-wide information system designed to coordinate all the resources, information, and activities needed to complete business processes such as order fulfillment or billing.

An ERP system supports most of the business system that maintains in a single database the data needed for a variety of business functions such as Manufacturing, Supply Chain Management, Financials, Projects, Human Resources and Customer Relationship Management.

An ERP system is based on a common database and a modular software design. The common database can allow every department of a business to store and retrieve information in real-time. The information should be reliable, accessible, and easily shared. The modular software design should mean a business can select the modules they need, mix and match modules from different vendors, and add new modules of their own to improve business performance.

Ideally, the data for the various business functions are integrated. In practice the ERP system may comprise a set of discrete applications, each maintaining a discrete data store within one physical database.

The term ERP originally referred to how a large organization planned to use organizational wide resources. In the past, ERP systems were used in larger more industrial types of companies. However, the use of ERP has changed and is extremely comprehensive, today the term can refer to any type of company, no matter what industry it falls in. In fact, ERP systems are used in almost any type of organization – large or small.

In order for a software system to be considered ERP, it must provide an organization with functionality for two or more systems. While some ERP packages exist that only cover two functions for an organization (QuickBooks: Payroll & Accounting), most ERP systems cover several functions.

Today’s ERP systems can cover a wide range of functions and integrate them into one unified database. For instance, functions such as Human Resources, Supply Chain Management, Customer Relations Management, Financials, Manufacturing functions and Warehouse Management functions were all once stand alone software applications, usually housed with their own database and network, today, they can all fit under one umbrella – the ERP system.

ERP – Implementation is The Challenge

The Ideal ERP System

An ideal ERP system is when a single database is utilized and contains all data for various software modules. These software modules can include:

Manufacturing: Some of the functions include; engineering, capacity, workflow management, quality control, bills of material, manufacturing process, etc.

Financials: Accounts payable, accounts receivable, fixed assets, general ledger and cash management, etc.

Human Resources: Benefits, training, payroll, time and attendance, etc

Supply Chain Management: Inventory, supply chain planning, supplier scheduling, claim processing, order entry, purchasing, etc.

Projects: Costing, billing, activity management, time and expense, etc.

Q.2.
A.2.

Introduction

When there are many projects run by an organization, it is vital for the organization to manage their project portfolio. This helps the organization to categorize the projects and align the projects with their organizational goals.

Project Portfolio Management (PPM) is a management process with the help of methods aimed at helping the organization to acquire information and sort out projects according to a set of criteria.

Objectives of Project Portfolio Management

Same as with financial portfolio management, the project portfolio management also has its own set of objectives. These objectives are designed to bring about expected results through coherent team players.

When it comes to the objectives, the following factors need to be outlined.

·        The need to create a descriptive document, which contains vital information such as name of project, estimated timeframe, cost and business objectives.

·        The project needs to be evaluated on a regular basis to ensure that the project is meeting its target and stays in its course.

·        Selection of the team players, who will work towards achieving the project's objectives.

Benefits of Project Portfolio Management

Project portfolio management ensures that projects have a set of objectives, which when followed brings about the expected results. Furthermore, PPM can be used to bring out changes to the organization which will create a flexible structure within the organization in terms of project execution. In this manner, the change will not be a threat for the organization.

The following benefits can be gained through efficient project portfolio management:

·        Greater adaptability towards change.

·        Constant review and close monitoring brings about a higher return.

·        Management's perspectives with regards to project portfolio management is seen as an 'initiative towards higher return'. Therefore, this will not be considered to be a detrimental factor to work.

·        Identification of dependencies is easier to identify. This will eliminate some inefficiency from occurring.

·        Advantage over other competitors (competitive advantage).

·        Helps to concentrate on the strategies, which will help to achieve the targets rather than focusing on the project itself.

·        The responsibilities of IT is focused on part of the business rather than scattering across several.

·        The mix of both IT and business projects are seen as contributors to achieving the organizational objectives.

Q.3.
A.3.

To facilitate the management decision making at all levels of company, the MIS must be integrated. MIS units are company wide. MIS is available for the Top management. Thetop management of company should play an active role indesigning, modifying and maintenance of the total organization wide management information system.Information system and Information technology have become a vital component of any successful business and are regarded as major functional areas just like any other functional area ofa business organization like marketing, finance, productionand HR. Thus it is important to understand the area of information system just like any other functional area in the business. MIS is important because all businesses have a needfor information about the tasks which are to be performed.Information and technology is used as a tool for solvingproblems and providing opportunities for increasing productivity and quality.Information has always been important but it has never been so available, so current and so overwhelming. Efforts have been made for collection and retrieval of information,However, challenges still remain in the selection analysis and interpretation of the information that will further improve decision making and productivity.

MIS for a Business Organization :

a. Support the Business Process :

Treats inputs as a request from the customer and outputs as services tocustomer. Supports current operations and use the system to influence further way of working.

 b. Support Operation of a Business Organization:

MIS supports operations of a business organization by giving timely information, maintenance and enhancement which provides flexibility in the operation of an organizations.

c. To Support Decision Making :

MIS supports the decision making by employee in their daily operations. MIS also supports managers in decision making to meet the goa lsand objectives of the organization. Different mathematical models and IT tools are used for the purpose evolving strategies to meet competitive needs.

d. Strategies for an Organization :

Today each business is running in a competitive market. MIS supports the organization to evolve appropriate strategies for the business to assented in a competitive environment.The objective of an MIS (Management Information System) isto provide information for decision making on planning,initiating, organizing, and controlling the operations of the subsystems of the form and to provide a synergetic organization in the process. Decision Support System: It is sometimes described as the next evolutionary step after Management Information Systems (MIS) . MIS support decision making in both structured and unstructured problem environments.. It supports decision making at all levels of the organization .IS (Information Systems) are intended to be woven into the fabric of the organizations , not standing alone.IS support all aspects of the decision making process. MIS are made of people, computers, procedures, databases,interactive query facilities and so on. They are intended to be evolutionary/adaptive and easy for people to use. The human intelligence is closely related with the human experience and decision making skills which is strongly backed by information's. Now a day's in every field of human working right information is considered as the most important resource of good decision making. Every organization runs by the managers of organization, who are making decisions in every step of organizational activities. Due to the importance of information in decision making a separate field has emerged to serve the appropriate information's to managers for effective and good decision making purpose. Serving the suitable information use to pass through a process called management information system as the information is using to make management decisions
Q.4.
A.4.
The e-Business model, like any business model, describes how a company functions; how it provides a product or service, how it generates revenue, and how it will create and adapt to new markets and technologies. It has four traditional components as shown in the figure, The e-Business Model. These are the e-business concept, value proposition, sources of revenue, and the required activities, resources, and capabilities. In a successful business, all of its business model components work together in a cooperative and supportive fashion.

Although an e-Business is often thought of as e-Commerce, there are other types of online activities that fall under the definition of e-Business that can benefit from this discussion (see e-Business Basics for basic concepts and definitions).

E-Business Concept

The e-business concept describes the rationale of the business, its goals and vision, and products or offerings from which it will earn revenue. A successful concept is based on a market analysis that identifies customers likely to purchase the product and how much they are willing to pay for it.

Goals And Objectives

The e-Business concept should be based, in part, on goals such as "become a major car seller, bank, or other commercial enterprise", and "to become a competitor to some of the well-known firms in each of these industries." Objectives are more specific and measurable, such as "capture 10% of the market", or "have $100 million in revenues in five years." Whether these goals and objectives are realistic or not, and whether the company is prepared to achieve these goals is addressed in the business plan process for startup firms and in the implementation plan for an existing firm that is considering a significant change. In looking at the business model it is sufficient to know what the goals and objectives are, and whether they are being pursued.

Corporate Strategies

Embedded in the e-Business concept are strategies that describe how the business concept will be implemented. These are known as corporate strategies because they establish how the business is intended to function. These strategies can be modified to improve the performance of the business. Environmental strategies, discussed in a following section, describe how the company will address external environmental factors, over which it has no control.

The E-Business Concept And Market Research

The selection and refinement of the business concept should be integrally tied into knowledge of the market it serves. In performing market research care must be taken to account for the global reach of the Internet for both customers and competitors. It is also important to remember that markets shift, and can shift rapidly under certain conditions. But most important is to truly understand what the market is, who comprises it, and what do they want.

Figure: The E-Business Concept

Q.5.
A.5.

OLAP The term, of course, stands for ‘On-Line Analytical Processing’. But that is not only a definition; it’s not even a clear description of what OLAP means. It certainly gives no indication of why you would want to use an OLAP tool, or even what an OLAP tool actually does. And it gives you no help in deciding if a product is an OLAP tool or not.
We hit this problem as soon as we started researching The OLAP Report in late 1994 as we needed to decide which products fell into the category. Deciding what is an OLAP has not got any easier since then, as more and more vendors claim to have ‘OLAP compliant’ products, whatever that may mean (often they don’t even know). It is not possible to rely on the vendors’ own descriptions and membership of the long-defunct OLAP Council was not a reliable indicator of whether or not a company produces OLAP products. For example, several significant OLAP vendors were never members or resigned, and several members were not OLAP vendors. Membership of the instantly moribund replacement Analytical Solutions Forum was even less of a guide, as it was intended to include non-OLAP vendors.
The Codd rules also turned out to be an unsuitable way of detecting ‘OLAP compliance’, so we were forced to create our own definition. It had to be simple, memorable and product-independent, and the resulting definition is the ‘FASMI’ test. The key thing that all OLAP products have in common is multidimensionality, but that is not the only requirement for an OLAP product.
Online Analytical Processing, or OLAP is an approach to quickly provide answers to analytical queries that are multidimensional in nature. OLAP is part of the broader category business intelligence, which also encompasses relational reporting and data mining. The typical applications of OLAP are in business reporting for sales, marketing, management reporting, business process management (BPM), budgeting and forecasting, financial reporting and similar areas. The term OLAP was created as a slight modification of the traditional database term OLTP (Online Transaction Processing).
Databases configured for OLAP employ a multidimensional data model, allowing for complex analytical and ad-hoc queries with a rapid execution time. They borrow aspects of navigational databases and hierarchical databases that are speedier than their relational kin. Drill down in EIS
An Executive Information System (EIS) is a type of management information system intended to facilitate and support the information and decision making needs of senior executives by providing easy access to both internal and external information relevant to meeting the strategic goals of the organization. It is commonly considered as a specialized form of a Decision Support System (DSS).
The emphasis of EIS is on graphical displays and easy-to-use user interfaces. They offer strong reporting and drill-down capabilities. In general, EIS are enterprise-wide DSS that help top-level executives analyze, compare, and highlight trends in important variables so that they can monitor performance and identify opportunities and problems. EIS and data warehousing technologies are converging in the marketplace.
In recent years, the term EIS has lost popularity in favour of Business Intelligence (with the sub areas of reporting, analytics, and digital dashboards).
EIS enables executives to find those data according to user-defined criteria and promote information- based insight and understanding. Unlike a traditional management information system presentation, EIS can distinguish between vital and seldom-used data, and track different key critical activities for executives, both which are helpful in evaluate if the company is meeting its corporate objectives. After realizing its advantages, people have applied EIS in many areas, especially, in manufacturing, marketing,and finance areas.

Basically, manufacturing is the transformation of raw materials into finished goods for sale, or intermediate processes involving the production or finishing of semi-manufactures. It is a large branch of industry and of secondary production. Manufacturing operational control focuses on day-to-day operations, and the central idea of this process is effectiveness and efficiency. To produce meaningful
managerial and operational information for controlling manufacturing operations, the executive has to make changes in the decision processes. EIS provides the evaluation of vendors and buyers, the evaluation of purchased materials and parts, and analysis of critical purchasing areas. Therefore, the executive can oversee and review purchasing operations effectively with EIS. In addition, because production planning and control depends heavily on the plant’s data base and its communications with all manufacturing work centers, EIS also provides an approach to improve production planning and control.
The future of executive info systems will not be bound by mainframe computer systems. This trend allows executives escaping from learning different computer operating systems and substantially decreases the implementation costs for companies. Because utilizing existing software applications lies in this trend, executives will also eliminate the need to learn a new or special language for the EIS package. Future executive information systems will not only provide a system that supports senior executives, but also contain the information needs for middle managers. The future executive information systems will become diverse because of integrating potential new applications and technology into the systems, such as incorporating artificial intelligence (AI) and integrating multimedia characteristics and ISDN technology into an EIS.

In tandem with the growth of the Internet and e-business, the number of digital data sources has increased immensely. These data sources contain important transactional data and are generally interconnected via a network. This has created a pressing need for a suitable executive information system (EIS) that is capable of extracting data from internal and external data sources and providing data analysis on demand for business executives. On-demand data analysis requires an information integration approach that can manage rapid changes in data sources. Existing EISs commonly adopt data warehousing technology to consolidate data from multiple sources in a tailor-made fashion, and support predefined multidimensional data analysis. However, this architecture is neither adaptable to changes in local sources nor flexible enough for ad hoc analyses. This paper develops methods and algorithms for a new EIS architecture that takes advantage of a meta-database to achieve adaptability and flexibility. A PC-based prototype is built to prove the concept.

Q.6.
A.6.

Advantages and disadvantages of knowledge management

Consider the measurable benefits of capturing and using knowledge more effectively in your business. The following are all possible outcomes:

·         An improvement in the goods or services you offer and the processes that you use to sell them. For example, identifying market trends before they happen might enable you to offer products and services to customers before your competitors.

·         Increased customer satisfaction because you have a greater understanding of their requirements through feedback from customer communications.

·         An increase in the quality of your suppliers, resulting from better awareness of what customers want and what your staff require.

·         Improved staff productivity, because employees are able to benefit from colleagues' knowledge and expertise to find out the best way to get things done. They'll also feel more appreciated in a business where their ideas are listened to.

·         Increased business efficiency, by making better use of in-house expertise.

·         Better recruitment and staffing policies. For instance, if you have increased knowledge of what your customers are looking for, you're better able to find the right staff to serve them.

·         The ability to sell or license your knowledge to others. You may be able to use your knowledge and expertise in an advisory or consultancy capacity. In order to do so, though, make sure that you protect your intellectual property. See protecting intellectual property.

Challenges of knowledge management

In order to maximise the benefit of knowledge management within your business you may have to overcome the following challenges:

·         Capturing and recording business knowledge - ensure your business has processes in place to capture and record business knowledge.

·         Sharing information and knowledge – develop a culture within your business for sharing knowledge between employees.

·         Business strategy and goals – without clear goals or a business strategy in place for the knowledge gathered the information will be of no use to your business.

·         Knowledge management systems – these systems can be costly and complex to understand but when utilised properly can provide huge business benefits. It is important that staff are fully trained on these systems so that they collect and record the right data.

Q.7.
A.7.

Data security should be an important area of concern for every small-business owner. When you consider all the important data you store virtually -- from financial records, to customers' private information -- it's not hard to see why one breach could seriously damage your business.

According to the most recent Verizon Data Breach Investigations Report [PDF], an estimated "285 million records were compromised in 2008." And 74 percent of those incidents were from outside sources.

We consulted Roland Cloutier, Chief Security Officer for ADP and a board member for the National Cyber Security Alliance, and Matt Watchinski, Senior Director of the Vulnerability Research Team for cyber security provider Source fire, to find out the key security measures every small business should be taking.

1. Establish strong passwords
Implementing strong passwords is the easiest thing you can do to strengthen your security.

Cloutier shares his tip for crafting a hard-to-crack password: use a combination of capital and lower-case letters, numbers and symbols and make it 8 to 12 characters long.

According to Microsoft, you should definitely avoid using: any personal data (such as your birthdate), common words spelled backwards and sequences of characters or numbers, or those that are close together on the keyboard.

As for how often you should change your password, Cloutier says that the industry standard is "every 90 days," but don't hesitate to do it more frequently if your data is highly-sensitive.

Another key: make sure every individual has their own username and password for any login system, from desktops to your CMS. "Never just use one shared password," says Cloutier.

And finally, "Never write it down!" he adds.

2. Put up a strong firewall
In order to have a properly protected network, "firewalls are a must," Cloutier says.

A firewall protects your network by controlling internet traffic coming into and flowing out of your business. They're pretty standard across the board -- Cloutier recommends any of the major brands.

3. Install antivirus protection
Antivirus and anti-malware software are essentials in your arsenal of online security weapons, as well.

"They're the last line of defense" should an unwanted attack get through to your network, Cloutier explains.

4. Update your programs regularly
Making sure your computer is "properly patched and updated" is a necessary step towards being fully protected; there's little point in installing all this great software if you're not going to maintain it right.

"Your security applications are only as good as their most recent update," Watchinski explains. "While applications are not 100 percent fool-proof, it is important to regularly update these tools to help keep your users safe."

Frequently updating your programs keeps you up-to-date on any recent issues or holes that programmers have fixed.

5. Secure your laptops
Because of their portable nature, laptops are at a higher risk of being lost or stolen than average company desktops. It's important to take some extra steps to make certain your sensitive data is protected.

Cloutier mandates "absolutely: encrypt your laptop. It's the easiest thing to do."

Encryption software changes the way information looks on the harddrive so that, without the correct password, it can't be read.

Cloutier also stresses the importance of never, ever leaving your laptop in your car, where it's an easy target for thieves. If you must, lock it in your trunk.

6. Secure your mobile phones
Cloutier points out that smartphones hold so much data these days that you should consider them almost as valuable as company computers -- and they're much more easily lost or stolen. As such, securing them is another must.

Q.8.

A.8.

Total cost of ownership (TCO) is an analysis that places a single value on the complete life cycle of a capital purchase. This value includes every phase of ownership: acquisition, operation, and the softer costs of change management that flows down from acquisition such as documentation and training.

What is included in TCO Analysis?

There are three key components to TCO calculations:

1.    Acquisition/Physical Hardware Costs

2.    Operating Costs

3.    Personnel Costs

Let’s look at each of these in turn.

Acquisition Costs

Acquisition/Physical Hardware costs include the cost of equipment or property before taxes, but after commissions, discounts, purchasing incentives, and closing costs. Sometimes this will include one-time peripheral equipment or upgrades necessary to installation or utilization of the asset.

Operating Costs

Operating costs include subscriptions or services needed to put the item into business use. This includes utility costs, direct operator labor, and initial training costs.

Personnel Costs

Personnel overhead may include administrative staffing, support personnel to the equipment, facility housing the equipment and operators. This may include ongoing training and troubleshooting labor for maintenance purposes.

Accounting Contributions

True total cost can include not only costs but incremental savings or revenue flows created by the capital investment. The change in cash flows versus the "business as usual" option is what mitigates total cost of ownership (TCO). Those monies must be valued using Net Present Value calculations to consider the values over time.

Real total cost of ownership (TCO) analysis is a critical tool in the decision-making toolbox for any sized business. It requires both an understanding of the investment considered and the potential business impact to find the right answer.

MCA 1st sem /MCS-013/Solved Assignment/ Discrete Mathematics /2017-2018 New

Question.1.(A)

Ans a . Logical Connectivity:- 

In logic, a logical connective (also called a logical operator) is a symbol or word used to connect two or more sentences (of either a formal or a natural language) in a grammatically valid way, such that the sense of the compound sentence produced depends only on the original sentences. The most common logical connectives are binary connectives (also called dyadic connectives) which join two sentences which can be thought of as the function's operands. Also commonly, negation is considered to be a unary connective.

Different Types of Logical Connectivity:-

 Disjunction :-

Definition: The disjunction of two propositions p and q is the compound statement p or q, denoted by p ∨ q.

For example, ‘Zarina has written a book or Singh has written a book.’ Is the

disjunction of p and q, where

p : Zarina has written a book, and

q : Singh has written a book.

Similarly, if p denotes ‘ 2 > 0’ and q denotes ‘2 < 5’, then p ∨ q denotes the statement

‘2 is greater than 0 or 2 is less than 5.’.

Let us consider an example.

Example 1: Obtain the truth value of the disjunction of ‘The earth is flat’.

and ‘3 + 5 = 2’.

Solution: Let p denote ‘The earth is flat,’ and q denote ‘3 + 5 = 2’. Then we know

that the truth values of both p and q are F. Therefore, the truth value of p ∨ q is F.

Conjunction:-

As in ordinary language, we use ‘and’ to combine simple propositions to make

compound ones. For instance, ‘ 1 + 4 ≠ 5 and Prof. Rao teaches Chemistry.’ is

formed by joining ‘1 + 4 ≠ 5’ and ‘Prof. Rao teaches Chemistry’ by ‘and’. Let us

define the formal terminology for such a compound statement.

Definition: 

We call the compound statement ‘p and q’ the conjunction of the

statements p and q. We denote this by p ∧ q.

For instance, ‘3 + 1 ≠ 7 ∧ 2 > 0’ is the conjunction of ‘3 + 1 ≠ 7’ and ‘2 > 0’.

Similarly, ‘2 + 1 = 3 ∧ 3 = 5’ is the conjunction of ‘2 + 1 = 3’ and ‘3 = 5’.

Now, when would p ∧ q be true? Do you agree that this could happen only when both

p and q are true, and not otherwise? For instance, ‘2 + 1 = 3 ∧ 3 = 5’ is not true

because ‘3 = 5’ is false.

So, the truth table for conjunction.

Negation:-

Definition: The negation of a proposition p is ‘not p’, denoted by ~p.

For example, if p is ‘Dolly is at the study center.’, then ~ p is ‘Dolly is not at the study

center’. Similarly, if p is ‘No person can live without oxygen.’, ~ p is ‘At least one

person can live without oxygen.’.

Conditional Connectives:-

Consider the proposition ‘If Shruti gets 75% or more in the examination, then she

will get an A grade for the course.’. We can write this statement as ‘If p, and q’,

where

p: Shruti gets 75% or more in the examination, and

q: Shruti will get an A grade for the course.

This compound statement is an example of the implication of q by p.

Definition: Given any two propositions p and q, we denote the statement ‘If p, then

q’ by p → q. We also read this as ‘p implies q’. or ‘p is sufficient for q’, or ‘p only if

q’. We also call p the hypothesis and q the conclusion. Further, a statement of the

form p → q is called a conditional statement or a conditional proposition.

Precedence Rule

While dealing with operations on numbers, you would have realized the need for

applying the BODMAS rule. According to this rule, when calculating the value of an

arithmetic expression, we first calculate the value of the Bracketed portion, then apply

Of, Division, Multiplication, Addition and Subtraction, in this order. While

calculating the truth value of compound propositions involving more than one

connective, we have a similar convention which tells us which connective to apply

first.

Why do we need such a convention? Suppose we didn’t have an order of preference,

and want to find the truth of, say ~ p ∨ q. Some of us may consider the value of ( ~

p) ∨ q, and some may consider ~ ( p ∨ q). The truth values can be different in

these cases. For instance, if p and q are both true, then ( ~ p) ∨ q is true, but ~ (

p ∨ q) is false. So, for the purpose of unambiguity, we agree to such an order or

rule.

The rule of precedence: The order of preference in which the connectives are

applied in a formula of propositions that has no brackets is

:-

i) ~

ii) ∧

iii) ∨ and ⊕

iv) → and ↔

Q.1.(b)

A.1(b)

i)  

  

  

ii)

Q.1.(c)

A.1.(c)

(i)

(ii)

Q.1.(d)

A.1.(d) Logical Equivalence:-

Logical equivalence is a type of relationship between two statements or sentences in propositional logic or Boolean algebra. The relation translates verbally into "if and only if" and is symbolized by a double-lined, double arrow pointing to the left and right ( ). If A and B represent statements, then A B means "A if and only if B."

The statement A B is exactly the same as

(A B) * (B A)

where the asterisk (*) represents the logical AND operation, and the right-pointing, double-lined arrow ( ) represents logical implication

Question.2.(A)

Ans a . 

(i) (∃x) ( ∀y) ( ∃z) P 

(ii) ( ∀x) ( ∃y) (∃ z) P 

(B)

Ans b.

 (i)  Some students can pass in exam. 

∀x(C(x)∧P(x))

Then you say that everything in the domain is a student in the course and passed the exam.

However, presumably there are things in the domain other than students in the course. For example, that book that the student didn't read, or the exam that they took. So if they are in the domain, you would get that the book passed the exam, or that the exam is a student in the course ... which is not what you want. What you want is that if something is a student in the course, then they passed the exam, i.e. you want the conditional.

Of course, it is possible that the domain is simply all the students in the course, but then why would you need a predicate C(x)C(x)? If the domain was all students in the course, you could simply use ∃x¬B(x)∃x¬B(x) for the first premise, and ∀xP(x)∀xP(x) for the second.

So, either way, ∀(C(x)∧P(x))

∀(C(x)∧P(x)) is not what you want.

(ii) Everything is having life. 

∀x(L(x)∧ E(x))

We can say that Everything is having Life. 

and In another way , we can say that, 

~ (∀x(C(x)∧P(x))), that 's mean Nothing is having Life.

(C)

Ans c.

Definition Of Indirect Proof

Indirect proof is a type of proof in which a statement to be proved is assumed false and if the assumption leads to an impossibility, then the statement assumed false has been proved to be true.

Example of Indirect Proof

Sum of 2n even numbers is even, where n > 0. Prove the statement using an indirect proof. 

The first step of an indirect proof is to assume that 'Sum of even integers is odd.' 

That is, 2 + 4 + 6 + 8 + . . . .+ 2n = an odd number

⇒ 2(1 + 2 + 3 + 4 + . . . + n) = an odd number ⇒ 2 × = an odd number

⇒ n(n + 1) = an odd number, a contradiction, because n(n + 1) is always an even number. 

Thus, the statement is proved using an indirect proof.

(D)

Ans d.

An equivalence relation on a set S, is a relation on S which is reflexive, symmetric and transitive. Examples: Let S = ℤ and define R = {(x,y) | x and y have the same parity} i.e., x and y are either both even or both odd. The parity relation is an equivalence relation. 1. For any x ∈ ℤ, x has the same parity as itself, so (x,x) ∈ R. 2. If (x,y) ∈ R, x and y have the same parity, so (y,x) ∈ R. 3. If (x,y) ∈ R, and (y,z) ∈ R, then x and z have the same parity as y, so they have the same parity as each other (if y is odd, both x and z are odd; if y is even, both x and z are even), thus (x,z) ∈ R.

FOR EXAMPLE

Let S = ℝ and define the "square" relation R = {(x,y) | x 2 = y 2}. The square relation is an equivalence relation. 1. For all x ∈ ℝ, x 2 = x 2 , so (x,x) ∈ R. 2. If (x,y) ∈ R, x 2 = y 2 , so y 2 = x 2 and (y,x) ∈ R. 3. If (x,y) ∈ R and (y,z) ∈ R then x 2 = y 2 = z2 , so (x,z) ∈ R. For any set S, the identity relation on S, I S = {(x,x) | x ∈ S}. This is an equivalence relation for rather trivial reasons. 1. obvious 2. If (x,y) ∈ R then y = x, so (y,x) = (x,x) ∈ R. 3. If (x,y) ∈ R and (y,z) ∈ R then x = y = z, so (x,z) = (x,x) ∈ R.

Question.3.

a)

b)

ANS:- 

c)Explain De Morgan’s laws in relation to Boolean Algebra.

 Ans .DeMorgan’s Theorem is mainly used to solve the various Boolean algebra expressions.The Demorgan’s theorem defines the uniformity between the gate with same inverted input and output. It is used for implementing the basic gate operation likes NAND gate and NOR gate. The Demorgan’s theorem mostly used in digital programming and for making digital circuit diagrams. There are two DeMorgan’s Theorems.

Theorem 1

o    The left hand side (LHS) of this theorem represents a NAND gate with inputs A and B, whereas the right hand side (RHS) of the theorem represents an OR gate with inverted inputs.

o    This OR gate is called as Bubbled OR.

Table showing verification of the De Morgan's first theorem −

Theorem 2

o    The LHS of this theorem represents a NOR gate with inputs A and B, whereas the RHS represents an AND gate with inverted inputs.

o    This AND gate is called as Bubbled AND.

Table showing verification of the De Morgan's second theorem −

(d) What is principle of mathematical induction? Explain with the help of an example.

Ans :- Mathematical induction is a mathematical proof technique used to prove a given statement about any well-ordered set. Most commonly, it is used to establish statements for the set of all natural numbers.

Mathematical induction is a form of direct proof, usually done in two steps. When trying to prove a given statement for a set of natural numbers, the first step, known as the base case, is to prove the given statement for the first natural number. The second step, known as the inductive step, is to prove that, if the statement is assumed to be true for any one natural number, then it must be true for the next natural number as well. Having proved these two steps, the rule of inference establishes the statement to be true for all natural numbers. In common terminology, using the stated approach is referred to as using the Principle of mathematical induction.

Example: 'The sum of the natural numbers from 1 to n equals n(n+1)/2'.

This is a property which depends on n. We write this fact as E(n). E(3) means in our example :

' The sum of the natural numbers from 1 to 3 equals 3(3+1)/2'.

A proof by induction is a common method to prove such a property. We show how the method works and why it is correct.

Let V = { n | E(n) is true }. First we prove E(1). Then we know that 1 is in V.

Then we prove that: If E(k) is true, then E(k+1) is true.

If that is proved, we know that: if k is in V then (k+1) is in V.

But 1 is in V, so 2 is in V, so 3 is in V, etc...

Then V = { 1,2,3,4,....}

We have shown that the property E(n) is true for all natural numbers starting with 1.

In our example is E(n) is 1+2+3+4+5+...+ n = (1+n)n/2

First step:

We'll show the truth of E(1). We have to show: 1 = (1+1).1/2

We see immediately that both sides are equal. So, E(1) is true.

V contains '1'.

Second step:

Assume for a moment that E (k) is true.

Given : 1+2+3+4+5+...+k = (1+k)k/2

We'll show the truth of E(k+1).

Proof:

    left side

  = (1+2+3+4+5+...+k)+ (k+1)

  = (1+k).k/2 + (k+1)

  = (k+1)(k+2)/2

    right side

  = (1+(k+1)).(k+1)/2

  = (2+k)(k+1)/2

We see that the left side is equal to the right side.

Thus if E (k) is true then E(k+1) is true.

If k is in V, then k+1 is in V.

But 1 is in V, so 2 is in V, so 3 is in V, etc..

So, V = { 1,2,3,4,....}

Conclusion: The property is proved for all natural numbers starting from 1.

Question 4

(a)  How many different committees can be formed of 12 professionals, each containing at least 2 Professors, at least 3 Lecturers and 3 Administrative Officers from a set of 5 Professors, 8 Lectures and 5 Administrative Officers.

(b)   

PROFESSOR                                         2

 Lecturers                                                 3 

Administrative Officers                          3                                                                  

SOLUTION IS

C(5,2)  + C(8,3)  +  C(5,3)

( 5X4)/(2X1) + (8X7X6)/(3X2X1) + (5X4X3)/(3X2X1)

10+56+10

76

 (b) There are two mutually exclusive events A and B with P(A) =0.7 and P(B) = 0.6. Find the probability of followings:

 i) A and B both occur

P(A) = 1-P(A)= 1-0.7=0.3
P(B) = 1-P(B)= 1-0.6=0.4

 ii) Both A and B does not occur 

P(A’)=1-P(A)=1-0.7=0.3

P(B’)=1-P(B)=1-0.6=0.4

iii) Either A or B does not occur

P(A’)=1-P(A)=1-0.7=0.3

P(B’)=1-P(B)=1-0.6=0.4

P(A’ OR B’) = P(A’) + P(B’) - P(A’ AND B’) =0.3 + 0.4 - 0.3 = 0.4

(c) What is set? Explain the basic properties of sets.

First we specify a common property among "things" (we define this word later) and then we gather up all the "things" that have this common property.

For example, the items you wear: shoes, socks, hat, shirt, pants, and so on.

I'm sure you could come up with at least a hundred.

This is known as a set.

Or another example is types of fingers. This set includes index, middle, ring, and pinky.

So it is just things grouped together with a certain property in common.

Sets are the fundamental property of mathematics. Now as a word of warning, sets, by themselves, seem pretty pointless. But it's only when we apply sets in different situations do they become the powerful building block of mathematics that they are.

Question 5

(a) How many words can be formed using letter of UNIVERSITY using each letter at most once?

 i) If each letter must be used,

{U,N,I,V,E,R,S,T,Y}

n=9

NO’S OF WORD IS = 9!

9!= 7X6X5X4X3X2X1= 363880 WORD ( in the UNIVERSITY letter I two time be we count once time)

 ii) If some or all the letters may be omitted. 

=P(9,0) + p(9,1)+P(9,2) + p(9,3)+P(9,4) + p(9,5)+P(9,6) + p(9,7)

= 1 + 9 + (9x8) + (9x8x7) + (9x8x7x6) +(9x8x7x6x5) +(9x8x7x6x5x4) +(9x8x7x6x5x4x3)

=1+7+72 +504+3024 +15120 + 60480+181440

= 260648

b)Show using truth table that: ( p ® q) ® q Þ p Ú q

(c) Explain whether (p q) (q r) is a tautology or not.

(d) Explain addition theorem in probability

The addition theorem in the Probability concept is the process of determination of the probability that either event ‘A’ or event ‘B’ occurs or both occur. The notation between two events ‘A’ and ‘B’ the addition is denoted as '∪' and pronounced as Union.

The result of this addition theorem generally written using Set notation,

P (A ∪ B) = P(A) + P(B) – P(A ∩ B),

Where,

P (A) = probability of occurrence of event ‘A’

P (B) = probability of occurrence of event ‘B’

P (A ∪ B) = probability of occurrence of event ‘A’ or event ‘B’.

P (A ∩ B) = probability of occurrence of event ‘A’ or event ‘B’.

Addition theorem probability can be defined and proved as follows:

Let ‘A’ and ‘B’ are Subsets of a finite non empty set ‘S’ then according to the addition rule

P (A ∪ B) = P (A) + P (B) – P (A). P (B),

On dividing both sides by P(S), we get

P (A ∪ B) / P(S) = P (A) / P(S) + P (B) / P(S) – P (A ∩ B) / P(S) (1).

If the events ‘A’ and ‘B’ correspond to the two events ‘A’ and ‘B’ of a random experiment and if the set ‘S’ corresponds to the Sample Space ‘S’ of the experiment then the equation (1) becomes

P (A ∪ B) = P (A) + P (B) – P (A). P (B),

This equation is known as the addition theorem in probability.

Here the event A ∪ B refers to the meaning that either event ‘A’ or event ‘B’ occurs or both may occur simultaneously.

If two events A and B are Mutually Exclusive Events then A ∩ B = ф,

Therefore

P (A ∪ B) = P (A) + P(B) [since P(A ∩ B) = 0],

In language of set theory A ∩ B̅ is same as A / B.

(e) Prove that the inverse of one-one onto mapping is unique.

A function is onto if and only if for every yy in the codomain, there is an xx in the domain such that f(x)=yf(x)=y.

So in the example you give, f:R→R,f(x)=5x+2f:R→R,f(x)=5x+2, the domain and codomain are the same set: R.R. Since, for every real number y∈R,y∈R, there is an x∈Rx∈R such that f(x)=yf(x)=y, the function is onto. The example you include shows an explicit way to determine which xx maps to a particular yy, by solving for xx in terms of y.y. That way, we can pick any yy, solve for f′(y)=xf′(y)=x, and know the value of xx which the original function maps to that yy.

Side note:

Note that f′(y)=f−1(x)f′(y)=f−1(x) when we swap variables. We are guaranteed that every function ff that is onto and one-to-one has an inverse f−1f−1, a function such that f(f−1(x))=f−1(f(x))=xf(f−1(x))=f−1(f(x))=x.

Question 6

(a) How many ways are there to distribute 15 district objects into 5 distinct boxes with

 i) At least three empty box.

C(5,3) X P(15,2)X2¹³  +  C(5,4) X P(15,1)X1¹⁴  + C(5,5) X P(15,0)X0¹⁵

ii) No empty box. 

C(15,5) x 5⁵ x5!

(b) Explain principle of multiplication with an example. 

ANS:-Multiplication Principle states: If an event occurs in m ways and another event occurs independently in n ways, then the two events can occur in m × n ways.

Examples of Multiplication Principle

A pizza corner sells pizza in 3 sizes with 3 different toppings. If Bob wants to pick one pizza with one topping, there is a possibility of 9 combinations as the total number of outcomes is equal to Number of sizes of pizza × Number of different toppings.

(c)  Set A,B and C are: A = {1, 2, 3,5, 8, 11 12,13}, B = { 1,2, 3 ,4, 5,6 } and C={ 7,8,12, 13}. Find A B C , A B C, A B C and (B~C) 

A∩ B U C ={1,2,3,5,7,8,12,13}

A U B U C={ 1,2,3,4,5,6,7,8,11,12,13}

A U B ∩ C= {8,12,13}

(B~C)= { 1,2, 3 ,4, 5,6 }

(d) In a class of 40 students; 30 have taken science; 20 have taken mathematics and 8 has neither taken mathematic nor science. Find how many students have taken:

 i) both subjects.

Ans :-  

(i)

 Science(s) = 30

 Math(m) = 20

 and 8 student has neither taken mathematic nor science

Then ,

n(S) +n(M) - n(Total student) 

= 30+20 - 40

=10

ii) exactly one subject

Ans :- 20+10 = 30

Question 7 

(a) What is power set? Write power set of set A={1,2,5,6,7,9}.  

Ans (a)  {0}, {1}, {2}, {3}, {0,1}, {0,2}, {0,3} , {1,2}, {1,3}, {2,3}, {0,1,2}, {0,1,3},{1,2,3}, {0,2,3}, {0,1,2,3}

 P = {{}, {0}, {1}, {2}, {3}, {0,1}, {0,2}, {0,3} , {1,2}, {1,3}, {2,3}, {0,1,2}, {0,1,3},{1,2,3}, {0,2,3}, {0,1,2,3}}...{1,2,5,6,7},{2,5,6,7,9}........

Like that, the possible Power Set A = 64

(b) Draw truth table for (P®Q) n (Q®P) and explain whether it is a tautology or not

Ans (b) Same as above example .. Q5(c)

(c) What is a function? Explain domain and range in context of function, with the help of example 

Ans (c) 

Domain

The domain of a function is the complete set of possible values of the independent variable.

In plain English, this definition means:

The domain is the set of all possible x-values which will make the function "work", and will output real y-values.

When finding the domain, remember:

The denominator (bottom) of a fraction cannot be zero

The number under a square root sign must be positive in this section

Example 1 Here is the graph of \displaystyle{y}=\sqrt{{{x}+{4}}}

y= x+4:

(d) State and prove the Pigeonhole principle.

Ans (d) 

Suppose that a flock of 20 pigeons flies into a set of 19 pigeonholes to roost. Because there are
20 pigeons but only 19 pigeonholes, a least one of these 19 pigeonholes must have at least two
pigeons in it. To see why this is true, note that if each pigeonhole had at most one pigeon in it,
at most 19 pigeons, one per hole, could be accommodated. This illustrates a general principle
called the pigeonhole principle, which states that if there are more pigeons than pigeonholes,
then there must be at least one pigeonhole with at least two pigeons in it.

Theorem:

I) If “A” is the average number of pigeons per hole, where A is not an integer then

• At least one pigeon hole contains ceil[A] (smallest integer greater than or equal to A) pigeons
• Remaining pigeon holes contains at most floor[A] (largest integer less than or equal to A) pigeons

Or

II) We can say as, if n + 1 objects are put into n boxes, then at least one box contains two or more objects.
The abstract formulation of the principle: Let X and Y be finite sets and let f: X –> Y be a function.

• If X has more elements than Y, then f is not one-to-one.
• If X and Y have the same number of elements and f is onto, then f is one-to-one.
• If X and Y have the same number of elements and f is one-to-one, then f is onto.

Pigeonhole principle is one of the simplest but most useful ideas in mathematics. We will see more applications that proof of this theorem.

Example 1: If (Kn+1) pigeons are kept in n pigeon holes where K is a positive integer, what is the average no. of pigeons per pigeon hole?

Solution: average number of pigeons per hole = (Kn+1)/n
= K + 1/n
Therefore at least a pigeonholes contains (K+1) pigeons i.e., ceil[K +1/n] and remaining contain at most K i.e., floor[k+1/n] pigeons.
i.e., the minimum number of pigeons required to ensure that at least one pigeon hole contains (K+1) pigeons is (Kn+1).

Question 8 

(a)Find inverse of the following functions.

Ans (a)