A formal method of applying uncertainty to numbers and arithmetic, related to fuzzy logic.

Fuzzy numbers are sets of continuous real numbers. The conventional definition of a set defines certain objects to be inside the set while everything else is outside the set. A fuzzy set, however, defines a membership function ranging from 0 to 1 that states the degree to which any object belongs in the set. Thus, if we wanted to represent the set of tall people, Danny DeVito might have a membership value of 0.0, Kareem Abdul-Jabbar might have a membership value of 1.0, and I would likely fall somewhere between. We can define an alpha-cut on a fuzzy set to be the members whose membership function is greater than alpha; thus, we can decide what "tall" means by picking an alpha to suit us.

A fuzzy number is a fuzzy set of real numbers with a few restrictions:

• At least one element has a membership function of 1
• The membership function is continuous
• The membership increases monotonically as it approaches "1" from either side.

The membership function for a fuzzy number can look like a triangle, a trapezoid, a gaussian distribution, etc. If we pick some number alpha between 0 and 1 as a minimally acceptible membership value, then we can refer to all fuzzy numbers as intervals whose members have values greater than alpha. (This is termed an alpha-cut.) For example, the fuzzy number [6,8] could be represented with a triangular function centered at 7 and an alpha-cut at 0.5:

```
1.0 - - - - - - - - - - -+- - - - - - - - - -
/ \
/   \
/     \
0.5 - - - - - - - - -/- - - -\- - - - - - - -
/         \
/           \
/             \
0.0 -----------------------------------------
...  4  5   6   7   8  9  10  ...

```

Arithmetic functions on fuzzy numbers are defined as:

• [a,b] + [d,e] = [a+d,b+e]
• [a,b] - [d,e] = [a-e,b-d]