Something different

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 is a collection of objects, then a fuzzy set 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.