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Fuzzy Sets : Basics

January 24, 2008 in Mathematics | 1 comment

This is an introduction to fuzzy sets and set operations and not a comprehensive tutorial.

I came across fuzzy sets and set operations while doing some project and wanted to share a few basics things – I still have a lot to learn about fuzzy logic.

Fuzzy sets

Fuzzy sets have a degree of membership associated with each element in the set. For example, in the set of intelligent people, Albert Einstein could have a membership of .99, whereas just an above average guy could have a membership of .60. Basically, it does not have strict boundaries, in contrast to classical sets. For instance, in a classical set, the set of intelligent people will contain everybody with an IQ greater than a certain level and there would be no somewhat intelligent people; also, anyone even just one mark below the cut off will be considered not intelligent.

If X is a collection of objects, then a fuzzy set A can be defined as a set of ordered pairs:

where is called the membership function, which maps each element of X to a membership grade (or degree of membership) between 0 and 1. Membership grade of 1 means that element is fully included in the fuzzy set, whereas 0 means that element is not included. Fuzzy sets could be either discrete or continuous.

Fuzzy set operations
Subset

Union

Intersection

Complement

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Sexy Primes

July 30, 2007 in Mathematics | No comments

I have been going through some mathematics tutorials and books for the first mathematics exam at our university – errr…. the exam is tomorrow. Although it tests elementary stuff, they make it tricky to make it hard.

They ask these stupid questions like the number of primes between 70 and 110 or twin primes between 40 and 130. Some students take a list of all primes from 1 to 200 (it’s a open book exam) while some generate the list of primes during the exam.

While I was searching for a list of primes on the internet (I know, you could write a five line program to generate them – why bother?) I found this sexy thing – Sexy Primes. Two numbers p, p + 6 are called Sexy Primes if they both are prime numbers; for example, (5, 11), (7, 13), (11, 17) are sexy primes. I think the word sexy has been used because of six (p, p + 6).

However, luckily, we are not asked about this kind of primes in the examination.

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Are you beautiful?

July 14, 2007 in Mathematics | No comments

Well, there’s a mathematical explanation.

I found this website which gives a mathematical representation of “The Perfect Face”, while reading some articles about Mathematics.

This site is based on the studies of a plastic surgeon – Dr. Stephen Marquard. Similar studies have been done by ancient Greek scientists and Leonardo da Vinci.

The ratios between certain elements (known as Golden Elements) in a beautiful face are equal to the Golden Ratio. Golden ratio is the positive root of x2 – x – 1 = 0 which is approximately 1.618. For example, the ratio of the width of the mouth to the width of the cheek is 1.618.

BeautyAnalysis.com has facial masks and gives evidence by comparing it with beautiful faces. It also gives instructions on comparing a face with masks.

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