Fuzzy sets are defined as sets that contain elements having varying degrees of membership values. Given A and B are two fuzzy sets, here are the main properties of those fuzzy sets:

Commutativity :-

• (A ∪ B) = (B ∪ A)


• (A ∩ B) = (B ∩ A)

Associativity :-

• (A ∪ B) ∪ C = A ∪ (B ∪ C)


• (A ∩ B) ∩ C = A ∩ (B ∩ C)

Distributivity :-

• A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C)


• A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C)

Idempotent :-

• A ∪ A = A


• A ∩ A = A

Identity :-

• A ∪ Φ = A => A ∪ X = X


• A ∩ Φ = Φ => A ∩ X = A

Note: (1) Universal Set 'X' has elements with unity membership value.
(2) Null Set 'Φ' has all elements with zero membership value.

Transitivity :-

• If A ⊆ B, B ⊆ C, then A ⊆ C

Involution :-

• (A^c)^c = A

De morgan Property :-

• (A ∪ B)^c = A^c ∩ B^c


• (A ∩ B)^c = A^c ∪ B^c

Note: A ∪ A^c ≠ X ;  A ∩ A^c ≠ Φ

Since fuzzy sets can overlap "law of excluded middle" and "law of contradiction" does not hold good.

Continue Reading >>Properties of Fuzzy Sets

Given 'X' to be universe of discourse, A and B are two fuzzy sets with membership function μA(x) and μB(x) then,

Union

The union of two fuzzy sets A and B is a new fuzzy set A ∪ B also on 'X' with membership function defined as follow:

Intersection

Intersection of fuzzy sets A & B, is a new fuzzy set A ∩ B also on 'X' whose membership function is defined by

Compliment

Compliment of a fuzzy set A is A with membership function

Product of Two Fuzzy Sets

The product of two fuzzy sets A & B is a new fuzzy set A.B with membership function:

Equality

Two fuzzy sets A and B are said to be equal i.e, A = B if and only if μA(x) = μB(x) Which means their membership values must be equal.

Product of Fuzzy Sets with a Crisp Number

Multiplying a fuzzy set A by a crisp number 'n' results in a new fuzzy set n.A, whose membership function is

Power of a Fuzzy Set

The alpha power of a fuzzy set A is a new fuzzy set A^α whose membership function is:

that is, individual memberships power of α

Difference of Fuzzy Sets

The differences of two fuzzy sets A and B is a new fuzzy set A-B which is defined as

Disjunctive Sum of A & B

It is the new fuzzy set defined as follow:

Continue Reading >>Operations on Fuzzy Sets

Fuzzy set is a set containing the elements that have varying degree of membership. Fuzzy sets support a flexible sense of membership values of elements of a set.

The membership function μA(x) is associated with a fuzzy set A, such that the membership function maps every element of universal set 'X' to the interval [0,1] ↔ 0 ≤ μA(x) ≤ 1
It can also be written as μA(x) : X → [0, 1]

A fuzzy set can also be defined as follow:

"If X is universe of discourse, x is a particular element of 'X', then a fuzzy set A~ (tilde understroke) defined on X may be written as collection of ordered pairs:

The membership function μA(x) may be described by discrete values or by a continuous function.

Continue Reading >>Define Fuzzy Sets

Mapping is an important concept in relating the elements or subsets of one universal set to elements or sets in another universe of discourse. If X and Y are two different universal sets, and if element x is contained in 'X' corresponds to an element 1 contained in 'Y' then it is generally termed as a mapping from X to Y or as f : X -> Y

When A ={x, y, z}, mapping the characteristic function membership of elements is denoted by χ_A and is defined as: Membership of element x in set A χ_A = 1, x ∈ A
where x represents an element in set A.

Continue Reading >>Mapping of Classical Sets to Functions

Cloud computing has been used for years by leaders in IT to offer the ultimate flexible, mobile working solution. However it’s only recently that ISPs are starting to offer what they’ve dubbed as “Cloud Hosting”, to individuals and companies. So what exactly is virtual hosting?

Well, most technology-aware computer users have already used cloud computing for years and enjoy the ease-of-use and flexibility that remote web hosting and virtual server hosting allow them to use during their working day. They manage their whole working life including emails, and securely stored business files remotely from anywhere in the world. Being able to access your work files, contacts, presentations, web hosting, and video services from anywhere in the world with just a click of a button is the ideal solution for busy traveling business people.

Obviously it’s mainly business experts who appreciate the potential for cloud hosting, and working on the move, in recent hardware innovations like the fantastically successful iPad and iPhone. There’s no longer any need for old-fashioned USB or FireWire wires to connect to physical hard disks that you need to carry around with you.

A ‘cloud’, or virtual server hosting solution is more reliable than risking a HD failure or lost back up disk, all you need to a WiFi connection and the right ISP or web hosting service provider to store your files securely. Millions of people around the world, including IT experts can’t be wrong. But don’t worry if you’re ignorant to the potential advantages of cutting edge computer technology . These systems are extremely user friendly and easy-to-use for those less technically able in IT.

If you’re an expert in cutting-edge technology you’ll know how get the best out of the latest in WiFi computing potential. Streamline your productivity then you’ll definitely get the best out of your iPad or iPhone with virtual server hosting – as they say, "There’s a App for that!".

About the Author: Gillian Thompson is a professional online journalist with over 15 years of experience in writing about the IT industry. She has in-depth knowledge of Apple products, having spent 10 years in London working for Macworld Magazine. Now residing in sunny North Yorkshire, she’s enjoying a break from the daily challenges of maintaining email servers, Firewalls, web servers, and from providing front line technical support to both her colleagues and the ever critical magazine readers.

Continue Reading >>iPad App Perfect for Cloud Computing and Remote File Access

Not every business maintains a blog, but a growing number of its employees do and the last few years have highlighted the risk associated with mixing your personal opinions with your day job.

In 2008 the anonymous blogger "Troll Tracker" - who ran a blog devoted to speaking out against bad business practices at the company he worked for (Cisco) - was identified by his employer and now faces a defamation law suit. Other examples of this include airline worker Ellen Simonetti being fired for images she placed on her blog and most famously web designer Heather B. Armstrong (or "Dooce") who discussed her coworkers in her blog posts and was let go as a result of it.

Tips For Careful Blogging

1. Avoid discussing your work in detail.

One of the most common pieces of advice given by professional writers is to "Write what you know". If you plan on starting a blog, writing about your job seems like the best way to make sure you've always got something new to write about; but in order to keep yourself from drawing the attention of your company, try to stick to talking about your industry as a whole instead.

2. Check Your Company's Policy on Blogging

Not mentioning your job only goes so far. If your company has a strict code of conduct in relation to what can be said by its employees online, it's safe to say you may be running too great a risk discussing even your industry let alone your day to day work.

3. Stay off the company blog listing unless told to do so.

This may seem counter to what I've said so far, but blogging under the scrutinizing glare of your company's HR Czar is sure to make everything about blogging less enjoyable. To avoid this simply don't mention your blog to coworkers and never inquire as to whether your company has a list.

About the author: Arthur Czuma is an IT and web consultant for several Canadian hosting and Internet providers.

Continue Reading >>The Benefits And Risks Of Blogging About Your Job

Probability Mass Function

X is random variable x₁, x₂,....,x_n are its values and f₁, f₂,....,f_n are their corresponding frequencies and N is the total number of random experiment in sample space. The probability of x_i is denoted by P(x=x_i) and defined as P(x=x_i) = f_i / N, where

Now, the function P(x=x_i) defined as above is called the "Probability Mass Function"

Note :-

Continue Reading >>What is Probability Mass Function

Random Variable Definition

If S is the sample space of a random experiment and to each s_i belongs to S there corresponds unique real number X(s_i), then the real valued function s thus defined is called a random variable defined on S.

Values of a Random Variable

Suppose R is range of a random variable X and R = {x₁, x₂,.....,x_n}then (x₁, x₂,.....,x_n) are called values of the random variable X.

Frequency of a Random Variable

Suppose X is a random variable defined on a sample space S. If x₁, x₂,.....,x_n are values of S and if f_i simple events are corresponding to the same value X_i of X for i=1 to n, then f₁, f₂,......,f_n are called frequencies of x₁, x₂,.....,x_n respectively.

Note :-

S is a sample space of a random experiment. N is the number of simple events on S. x₁, x₂,.....,x_n are values of X and f₁, f₂,......,f_n are corresponding frequencies then

Continue Reading >>Random Variable - Its Value and Frequency

Originally stated by the Reverend Thomas Bayes, this Bayes' Theorem falls under probability theory and according to it, if E₁, E₂, E₂, ..........,E_n are mutually exclusive and exhaustive events and A is any event then

Proof :-

Since E₁, E₂, E₂, ..........,E_n are pairwise exclusive and exhaustive events,

Continue Reading >>What is Bayes Theorem

Here is the definition of mathematical expectation which is also called as frequency or average number of times of occurrence of an event.

Suppose a random experiment is conducted independently 'n' times.
E is an event with probability P,
then out of n times approximate number of times of occurrence of event E is called mathematical expectation or frequency of E.

If the probability of any event E is P, then the frequency of E = n.P

Odds in favor of a event E are defined as P(E) to P(E)

Odds against E are defined as P(E) to P(E)

Continue Reading >>What is Mathematical Expectation

Independent Events

Two events A, B are said to be independent if and only if the Probability of (A intersection B) equals Probability of A multiplied to Probability of B.
P(A∩B) = P(A). P(B)

Note that, if A,B are independent events then P(B/A) = P(B) and P(A/B) = P(A)

Three elements A,B,C are independent if and only if
P(A∩B∩C) = P(A). P(B). P(C)

The following statements are equivalent which means everyone of them implies the remaining three:

1. Two events A,B are independent.


2. Two events A, B are independent.


3. Two events A, B are independent.


4. Two events A, B are independent.

2 → 3
Suppose A, B are independent. This means P(A ∩ B) = P(A). P(B)

Consider P(A ∩ B) = P (B - A)

= P(B) - P(A ∩ B)

= P(B) - P(A) . P(B)

= P(B) (1 - P(A))

= P(A).P(B)

Therefore A, B are independent.

3 → 4
Suppose A, B are independent.
so, P(A∩B) = P(A).P(B)

P(A ∩ B) = P(A ∪ B)

= 1 - P(A ∪ B)

= 1 - P(A) - P(B) + P(A ∩ span style="text-decoration: overline">B)

= P(A) - P(B) + P(A) - P(B)

= P(A) - P(B) (1 - P(A))

= P(A) (1 - P(B))

= P(A). P(B)

Therefore A, B are independent.

4 → 1
Suppose A, B are independent.
so we have P(A ∩ B) = P(A) . P(B)

P(B - (A∩B)) = (1 - P(A)) . P(B)

P(B) - P(A∩B) = P(B) - P(A) . P(B)

P(A∩B) = P(A) . P(B)

Therefore A,B are independent.

Therefore we can conclude that the above four statements are equivalent.

Continue Reading >>Independent Events of Probability

Conditional Probability

If we are given that an event A is already happened then the probability of happening of another event B is called conditional probability, and is denoted by P(B/A)

Note that, for conditional probability A is called sample space.

Conditional Probability P(B/A) is as follows:
P(B/A) = (number of simple events in A∩B / total number of simple events in A)

Continue Reading >>What is Conditional Probability

Power Factor Meter

The power factor meter gives the power factor (p.f) angles which is nothing but the phase angles between the voltage and the current vectors and its scale is calibrated directly in terms of cosΦ. The construction of p.f meter is of different types of which dynamo-meter type is most used for electrical lab work. Dynamo-meter type p.f meter consists of fixed coil called as current coil (CC), moving coil called as pressure coil (PC) which moves in the field produced by the current coil carrying load current.

The power factor meter is tested by using "phantom loading" in order to reduce the power loss. In this method the current coil will carry the load current and it is supplied by a low voltage supply. Rated voltage is applied to the pressure coil (PC), through a phase shifting transformer. The purpose of this arrangement is to phase shift the phase angle between the PC voltage and the current in the current coil.

Here is the formula for finding '% error' at Unity Power Factor (UPF)

Continue Reading >>What is a Power Factor Meter

Capacitance Pickup:-

Capacitance Pickup is a transducer that converts angular displacement into electrical signal. It consists of ganged capacitance and is based on the principle of variation of effective area of conductors, when other parameters such as separation, distance and dielectric strength being kept constant.

The basis of angular displacement measurement with the help of capacitance pickup is frequency modulation system. Two sets of ganged identical condensers form a part of weinbridge oscillator. The frequency of oscillation f = 1/2πRC. If "C" is varied typically from 550pf to 50pf we get a frequency variation in the range of 110. The following figure represents the block diagram of capacitance pickup:

Procedure for finding characteristics of Capacitance Pickup:-

1. Connect the capacitance pickup cable to the input socket of the main unit.


2. Keep the input angular displacement to zero position.


3. Check the DPM readings for zero indication or obtain the same by adjusting pot P2.


4. Now, turn the shaft to fully anti-clockwise position and to obtain the DPM indication of 180° by operating knob P2 if necessary.


5. Note down the DPM indication for different input angular displacements of capacitance pickup.


6. Draw the graph of output versus input angular displacements. Its expected waveform is as given below:
   
   

The main precaution to be taken while conducting this study experiment is not to stretch the wire coming from capacitance sensor.

Continue Reading >>Characteristics of Capacitance Pickup

Hardware deals with the physically existing parts of a computer which we can touch and feel. Whereas software is the set of instructions given to a computer which we can see and feel.

Software is divided into 4 parts namely:

1. Operating System
2. Language
3. Package
4. Application

Operating System is an interface between the user and the computer.

Language is the collection of pre-defined words. Computer languages are of 3 types:

1. Low level languages which consists of only 0's and 1's


2. Middle or Assembly level languages which is a combination of low and high level languages.


3. High level languages which consists of purely English like words.

Package is developed from a language. It is used for some generalized particular purpose.

Application is developed from a package. It is for being applied. No further software can be developed from it.

Continue Reading >>What is Hardware and Software

A distribution which gives the probabilities of the values of a random variable function is called random variable distribution. If x₁, x₂,....,x_n are random values of X and P₁, P₂,....,P_n are their corresponding probabilities, then the random variable distribution of X can be written as follow:

X = xi	P(x=xi)
x1x2 . . . xn	P1 P2 . . . Pn

Continue Reading >>Random Variable Distribution

Fuzzy Propositions :-

Fuzzy propositions are assigned to fuzzy sets. Suppose a fuzzy proposition 'P' is assigned to a fuzzy set 'A', then the truth value of the proposition is proposed by T (P) = μ_A(x) where 0 ≤ μ_A(x) ≤ 1

Therefore truthness of a proposition P is membership value of x in fuzzy set A.

The logical connectives like disjunction, conjunction, negation and implication are also defined on fuzzy propositions.

Let, a fuzzy proposition 'P' is defined on a fuzzy set A
     Q is defined on fuzzy set B

Conjunction

P /\ Q : x is A and B

T( P /\ Q) = Min [ T(P), T(Q)]

Negation

T(P^c) = 1 - T(P)

Disjunction

P V Q : x in A or B

T (P V Q) = Max [ T(P), T(Q) ]

Implication

P → Q : x is A then x is B

T( P → Q ) = T (P^c V Q) = Max [ T(P^c, T(Q)]

If P is a proposition defined on set A on universe of discourse X and
Q is another proposition defined on set B on universe of discourse Y,
then the implication P → Q can be represented by the relation R

R = ( A X B) U (A^c X Y) = If A then B

If x ∈ A, where x ∈ X and A ⊂ X
then y ∈ B, where y ∈ Y and B ⊂ Y

Implication of Classical Logic:-

Properties P and Q are given by
P : x ∈ A, where A is defined on x.
Q : y ∈ B, where B is defined on y.

Then the implication P → Q is represented in set theoretic form by a relation R as

R = (A X B) U (A^c X Y)

The implication is equivalent to linguistic rule form, if x ∈ A then y ∈ B.
This relation as characteristic function is as follow


For the classical predicate logical rule, (P → Q) V (P^c → S) the linguistic rule form is,
if x is A then y is B, else y is C.
Where C is defined as
S : y is C , C ⊂ Y.

The above linguistic rule form is decomposed as if (x is A) then (y is B) or if (x is A) then (y is not B)

In set theoretic form it can be represented by the relation R = (A X B) U (A^c X C)

The characteristic function for above compound proposition is given by

For example, suppose we have two universe of discourse X and Y
X = {1, 2, 3, 4}
Y = {1, 2, 3, 4, 5, 6}

X is the universe of normalized temperatures.
Y is the universe of normalized pressures.

Two crisp sets A and B defined on universe of discourses X and Y are A={2, 3} ; B={3,4}

for the deductive inference if A and B, find the relational matrix R.

sets A and B in Zedah's notations are given by
A = {0/1 + 1/2 + 1/3 + 0/4}
B = {0/1 + 0/2 + 1/3 + 1/4 + 0/5 + 0/6}

for the deductive inference if A then B, the set theoretic form is given by the relation
R = (A X B) U (A^c X Y)

Continue Reading >>What are Fuzzy Propositions

Fujistu LifeBook PH520/1A comes as a new addition to Fujitsu's ever growing ultraportable PH-Series computer line-up.

True to its name, this ultraportable netbook weighs less than 1.4 kg and is easy enough to carry around.

LifeBook PH520/1A runs on Windows 7 Home Premium 32-bit Genuine OS and is powered by 1.70GHz AMD Athlon II Neo K125 Processor that is well backed up with AMD ATI Mobility Radeon HD 4225 Graphics, 2GB of RAM and about 320GB of Hard Disk Drive. For easy communication, LifeBook PH520/1A is equipped with Bluetooth wireless technology, Wi-Fi access, Ethernet and Wireless LAN (IEEE 802.11b/g/n compliant).

Fujitsu LifeBook PH520/1A is loaded with some other key features and specifications like:

• 11.6" LCD Display with a resolution of 1366×768 pixels


• Multi-card reader and HDMI port


• Three USB 2.0 ports


• Direct Memory Slot compatible with SD Card / Memory Stick


• Good Battery backup that promises approximately 6.2 hours


• Built-in frontal 1.3 megapixel web camera with integrated microphone and loudspeaker for easy audio video conferencing

The Fujitsu LifeBook PH520/1A Ultraportable Netbook is available for purchase in the Japanese market for approximately $870 and if you want to buy it online, then head over to @ Fujitsu Direct Sales WEB MART where it costs around $900 including tax.

Continue Reading >>Fujitsu LifeBook PH520/1A Overview

Aircel the fastest growing mobile phone service provider and a pioneer in making GPRS based data coupons popular in India has recently made a new addition to its Pocket Internet Service with the introduction of Pocket Internet Card that costs just 5 Rupees. As you may know, Aircel's Pocket Internet is a flexible service that allows Internet access on mobile as well as laptop or desktop PC.

Using this new Pocket Internet Card of Rs.5 you can enjoy 1 day browsing of 20 MB. You can also get free data usage of another 20 MB on the second day but that can be used only for accessing Hotmail.

Browsing validity starts on the day of activation and ends at midnight 12 am, which typically means it gets deactivated if not renewed after 1 day. And as usual activating this internet service is free and browsing is charged just @10p/10Kb after the free usage limit.

This pack also includes some free content downloads like exciting games, free wallpapers and a whole lot more which can be availed only through a specific WAP URL on mobile handsets.

Note: This is a limited time offer and is applicable till September 30, 2010.

Continue Reading >>Enjoy 1 Day Browsing of 20MB with Aircel New Pocket Internet Card of Rs 5

Binomial Distribution

This is also a random variable distribution. A random experiment is conducted n times independently. Suppose E is one event of the experiment, P is its probability. If each time the event occurs treat it as success otherwise treat it as failure then probability for success is equal to P.
Probability for failure (say) q = 1-p → p+q =1.

Now, out of the 'n' times number of success that we may get it 0 or 1 or 2 or..............n

Probability of getting r times success =

The random variable distribution in the values of random variable are the number of success is called Binomial Distribution.

Now, the binomial distribution can be written as follow:

Probability Distribution Function

Symbol :-

If x₁, x₂, x₃,........, x_n are values of a random variable X, then

for any two real number K₁, K₂.

for any real number K.

Definition :-

Probability distribution function of a random variable X is denoted by F and is defined on R as follow:

Properties :-

If F is an imaginary function, then

Theorems :-

1. The arithmetic mean of binomial variable X with parameters n, p is the product of n and p ie., n.p | here n is the number of times that the experiment is conducted and p is the probability of success.


2. Variance of a binomial variable X with parameters n, p is n.p.q where q = 1 - p

Continue Reading >>Discuss About Binomial Distribution