Are you young or old? And where is the boundary between young and old? Everybody has their view – it is highly subjective – and it is expressed in words, not numbers.

The notion of a classical (crisp) set is based on binary logic: an element either belongs to a set or doesn’t.

You either pass or fail a test.
A number is either even or odd.
You are either a member of a club or you aren’t.

The classical sets work well when you are dealing with objective and quantitative information. However, they are unable to succeed in representing subjective or qualitative concepts given in natural language, which are uncertain by nature (e.g., young, old).

This is where the fuzzy sets come to the rescue. The fuzzy set theory introduces linguistic variables. A linguistic variable is a variable whose values are words rather than numbers. Classical (binary) logic is based on classical sets. Similarly, fuzzy sets give rise to fuzzy logic.

The idea behind a fuzzy set is that of gradual and partial membership to a set without a sharp boundary (e.g., you don’t suddenly change from young to old). This idea is in tune with human language, judgement, and representation of reality.

Let’s look at the representation of young and old in a classical set and then a fuzzy set.

As illustrated in Figure 1 below, the membership of an element to a classical set is expressed by 1 (belongs) or 0 (doesn’t belong).

An example of the fuzzy set principle representing young and old.

Figure 1: Classical sets

The membership of an element to a fuzzy set is expressed via a membership function with values in the interval [0,1]. Note that the membership functions of the fuzzy sets can have different shapes.

An example of the fuzzy set principle representing young and old, now with a membership function value in the interval.

Figure 2: Fuzzy sets

Fuzzy logic is used in numerous fields:

Control systems engineering
Image processing
Power engineering
Industrial automation
Robotics
Consumer electronics
Optimisation

You will find fuzzy logic in numerous real life applications, including:

facial pattern recognition.
air conditioners, washing machines, and vacuum cleaners.
anti-skid braking systems.
control of subway systems and unmanned helicopters.
knowledge-based systems for multi-objective optimisation of power systems.
weather forecasting systems.
models for new product pricing or project risk assessment.
medical diagnosis and treatment plans.
stock trading.

Fuzziness