**1. Introduction**

A fuzzy number is a special fuzzy set, and the fact that its membership function is unimodal (although "rest stops" along the way up and down the mountain are allowed) and normal is particularly characteristic. The concept of fuzzy sets was introduced by L.A. Zadeh [1] in 1965. There are some definitions of it. We see that in Section 2.3. In each application of fuzzy numbers, you can decide which definition is most useful and choose it on a case-by-case basis. For example, [2] is a well-known reference that defines fuzzy numbers with an eye toward fuzzy analysis (See also [3–6]).

#### **1.1 Where should the concept of fuzzy numbers be used?**

The concept of fuzziness or fuzzy numbers appears when we try to distinguish between two or more things or measure something by sight or feeling. If their specific values were required, there would be no need to force fuzziness into the discussion.

For example, coefficients appearing in differential equations can be treated as fuzzy numbers. Let us consider the radiocarbon dating method. Equation describes how several samples of radioactive material decay

$$\frac{\text{dN}}{\text{dt}} = -\lambda \text{N}, \quad \lambda > 0,\tag{1}$$

was presented by E. Rutherford. Here *N* ¼ *N t*ð Þ is the number of atoms in a radioactive material at time *t*. Eq. (1) implies that the larger *λ*, the faster the sample decays. *λ* varies with the substance, of course, and is determined *experimentally* by the observer on a case-by-case basis (see, e.g., [7] for more information). This *λ* is set with some mathematical basis, but in some cases it may be determined empirically, or the accuracy of the observation equipment and the skill of the observer may be highly dependent on it. Therefore, it seems more realistic, appropriate and effective to put fuzziness in *λ* and treat it as a fuzzy number. Differential equations involving fuzzy numbers are generally called fuzzy differential equations. To find the fuzzy solution of a fuzzy differential equation, we can obtain the level-cut sets of the solution by considering its level-cut sets and solving the interval equation, so we can collect them over all levels. However, in order to do so, of course, the fuzzy differential equation must have a (fuzzy) solution. See, e.g., Refs. [3, 6] for the existence and uniqueness of solutions of fuzzy differential equations. Refs. [8–11] are also helpful in knowing how to solve specific fuzzy differential equations of the type as in Eq. (1). In particular, Ref. [8] covers elementary contents of fuzzy numbers and fuzzy differential equations.

#### **1.2 The aim of this chapter**

We treat not only fuzzy sets / numbers, but also "fuzzy-membership-grade fuzzy sets / numbers" in this chapter. These are usually called type-2 fuzzy sets / numbers. In order to distinguish fuzzy sets / numbers from these, they are called type-1 fuzzy sets / numbers with emphasis. These are defined and explained in Section 3. We can generally consider type-*n* fuzzy sets / numbers, but since type-2 fuzzy sets / numbers are used in practical applications, this chapter also deals exclusively with type-1 and type-2 fuzzy sets/numbers.

More precisely, a type-2 fuzzy set can be said to be a fuzzy set whose membership grades are (type-1) fuzzy numbers. For example, it is a fuzzy set such that the membership grade at which a room feels hot is "about 0.8." We can thus think that type-2 fuzzy theory is the application of type-1 fuzzy number theory.

Roughly speaking, the membership function of a type-2 fuzzy set is generally in the shape of a mountain standing above the base of a mountain type. This is because the base is the membership function with "width." The "width" represents the fuzziness of a membership grade (see **Figure 1**). The red curve is the membership function with zero fuzziness.

**Figure 1.** *The form of the "bottom" of a type-2 fuzzy set.*

#### *Review of Type-1 and Type-2 Fuzzy Numbers DOI: http://dx.doi.org/10.5772/intechopen.110495*

Type-2 fuzzy theory is applied mainly to represent linguistic variables in reasoning. In the first place, Zadeh [1] introduced the concept of type-2 fuzzy sets for this purpose in 1975. By introducing it, we can treat truth values such as "approximately true," "neither true nor false," "truth unknown," etc. This has greatly advanced the study of reasoning (see, e.g., [12–15]). In recent years, type-2 fuzzy number theory has been developing and is being studied in principle as well as in application. Applications to differential equation theory have also been made (see, e.g., [10, 11, 13, 16]. Type-1 fuzzy differential equation theory can be seen in Refs. [3, 6], etc. All of the above studies benefit from the concept of type-1 fuzzy numbers. Indeed, type-2 fuzzy theory is ultimately attributed to type-1 fuzzy theory because any type-2 fuzzy set is represented as a "coupling" of two type-1 fuzzy sets (See Section 3 for details), and hence, concrete computations are also done by level-cutting of type-1 fuzzy sets (see, e.g., [17] for operations for type-2 fuzzy sets). Moreover, the utility of the type-2 fuzzy concept will be explained in Section 3.5.

Therefore, it can be said to be important to interpret the concept of fuzzy numbers appropriately and consider fuzzy numbers whose level-cut sets are easily computed. From the above, we see the following in this chapter:

