Fuzzy set theory and its properties with examples

Fuzzy set is nothing but just a generalization of classical set theories. In this article you will get very preliminary concepts of fuzzy set theory.
You will be able to learn about alpha-cut, strong alpha-cut, support, core, normality fuzzy set, sub normality, cardinality and nucleus of a fuzzy set.
Also, you will find the very basic operation union, intersection and complement of fuzzy sets and explained with examples.
In general, a fuzzy set can be defined as combination of a crisp set and a membership function.

What is characteristic function of a crisp set?

Before proceeding to the basic concept of fuzzy set theory we need to know about the characteristic function of a crisp set.  The characteristic function of a crisp set can be defined as the assignment of values of either 1 or 0 to each element in the universal set, depending upon the condition that whether the element is included in that set or not.

The definition of a characteristic function can be written as follows:

Let $A$ be a set on a universal set$X$, the characteristic function ${\chi _A}:X \to \left\{ {0,1} \right\}$ is defined as

${\chi _A}(x) = 0,x \notin A;$

Or ${\mkern 1mu} {\chi _A}(x) = 1,x \in A$

In case of the crisp set an element either belongs to the set or does not belongs to the set. Therefore the inclusion in the set can be assigned as the value 1 and the opposite, i.e. exclusion can be assigned as 0.

Example of a characteristic function:

Consider $A = \left\{ {x,y,z} \right\}$ be a set on a universal set $X$.
So, by the definition of characteristic function

${\chi _A}(x) = 1$ since $x \in A$

Similarly, ${\chi _A}(y) = 1$ and ${\chi _A}(z) = 1$

But, ${\chi _A}(t) = 0$ since $t \notin A$

What is fuzzy set?

In 1965 Lotif Ali Zadeh introduced the concept of fuzzy set. Fuzzy set theory provides a mathematical frame work in which uncertainty may arises due to incomplete information about  any problem, or information which is not reliable, or due to the linguistic nature in which the problem is defined.
Fuzzy set theory is becoming a major area of study for implementation in decision making under uncertain environment, artificial intelligence, automated systems etc.
We can generalize the characteristic function in such a way that the value can be assigned to an elements of the universal set which fall within the range 0 and 1.
The generalized characteristic function is termed as the membership function. Larger value of membership function corresponding to an element will denote higher degree of inclusion in that set. Such set together with the membership function is called a fuzzy set.

Definition of a fuzzy set

A fuzzy set is one which assigns grades of membership between 0 and 1 to objects within its universe of discourse. If $X$ is a universal set then a fuzzy set A is defined by its membership function

${\mu _A}:X \to (0,1)$

An alternative definition of a fuzzy set is

Le $X$ be a universal set of discourse and $x$be any element of $X$ then a fuzzy set $A$ is defined on $X$can be written as a collection of ordered pairs

$A = \left\{ {(x,{\mu _A}(x)):x \in X} \right\}$

Example of fuzzy set

 $X = \left\{ {a,b,c} \right\}$ be a universal set, then a fuzzy set on $X$ can be denoted as

$A = \left\{ {(a,.7),(b,1),(c,.3)} \right\}$ or $A = \left\{ {\frac{a}{{.7}},\frac{b}{1},\frac{c}{{.3}},} \right\}$

This notation implies that the degree of inclusion of the elements  $a,{\rm{ }}b,{\rm{ }}c$  in the set is .7, 1, .3 respectively.


Like crisp sets, there are some terminologies associated with fuzzy sets which are given below:

$\alpha$-Cuts (alpha cut)of a fuzzy set

Consider a fuzzy set$A$ in$X$ and suppose $\alpha  \in \left( {0,1} \right)$ be any real number.  Then the $\alpha$-cut of$A$ , denoted by ${}^\alpha A$ and is defined as

${}^\alpha A = \left\{ {x:{\mu _A}(x) \ge \alpha } \right\}$

Example of $\alpha$-Cut 

$A = \left\{ {\left( {a,.6} \right),\left( {b,.5} \right),\left( {c,1} \right),\left( {d,.2} \right)} \right\}$ be any fuzzy set defined on a universal set $X = \left\{ {a,b,c,d} \right\}$.

Let $\alpha  = .5$, then

${}^\alpha A = \left\{ {x:{\mu _A}(x) \ge \alpha } \right\}$

$\Rightarrow {}^{.5}A = \left\{ {x:{\mu _A}(x) \ge .5} \right\}$

$\Rightarrow {}^{.5}A = \left\{ {a,b,c} \right\}$

Strong $\alpha$-Cut(strong alpha cut)of a fuzzy set

Consider a fuzzy set$A$ in$X$ and a real number$\alpha  \in \left( {0,1} \right)$. Then the strong $\alpha$-cut, denoted by ${}^{\alpha  + }A$ and is defined by

${}^{\alpha  + }A = \left\{ {x:X;{\mu _A}(x) > \alpha } \right\}$

Example of Strong $\alpha$-Cut

$A = \left\{ {\left( {a,.6} \right),\left( {b,.5} \right),\left( {c,1} \right),\left( {d,.2} \right)} \right\}$ be any fuzzy set defined on a universal set $X = \left\{ {a,b,c,d} \right\}$.

Let $\alpha  = .5$, then

${}^{\alpha  + }A = \left\{ {x:{\mu _A}(x) > \alpha } \right\}$

$\Rightarrow {}^{.5 + }A = \left\{ {x:{\mu _A}(x) > .5} \right\}$

$\Rightarrow {}^{.5 + }A = \left\{ {a,c} \right\}$

Support of a Fuzzy

The support of a fuzzy set $A$ is defined on $X$ as

$Support(A) = {}^{0 + }A = \left\{ {x:{\mu _A}(x) > 0} \right\}$

Core of a Fuzzy Set

The core of a fuzzy set$A$ is defined on $X$ is a crisp set defined as

$Core(A) = \left\{ {x:{\mu _A}(x) \ge 1} \right\} = {}^1A$

Example of Core of a fuzzy set

$A = \left\{ {\left( {a,.3} \right),\left( {b,.5} \right),\left( {c,1} \right),\left( {d,.2} \right)} \right\}$ be any fuzzy set defined on a universal set $X = \left\{ {a,b,c,d} \right\}$.

$Core(A) = \left\{ {x:{\mu _A}(x) \ge 1} \right\} = {}^1A = \left\{ c \right\}$

Height of a Fuzzy Set

The height of a fuzzy set $A$ is the largest membership grade obtained by any element in that set.

$h(A) = \mathop {\sup }\limits_{x \in X} \left\{ {{\mu _A}(x)} \right\}$

Example of Height 

Consider $A = \left\{ {\left( {a,.3} \right),\left( {b,.5} \right),\left( {c,0} \right),\left( {d,.8} \right)} \right\}$ be any fuzzy set defined on a universal set $X = \left\{ {a,b,c,d} \right\}$.

$h(A) = \mathop {\sup }\limits_{x \in X} \left\{ {{\mu _A}(x)} \right\} = \sup \left\{ {.3,.5,0,.8} \right\} = .8$

Normal fuzzy set

A fuzzy set $A$ defined on a set $X$ is said to be a normal fuzzy set if and only if
$h(A) = \mathop {\sup }\limits_{x \in X} \left\{ {{\mu _A}(x)} \right\}$for at least one $x \in X$
$A$is called subnormal fuzzy set if $h(A) < 1$.

Example of Normal fuzzy set

$A = \left\{ {\left( {a,.3} \right),\left( {b,.5} \right),\left( {c,1} \right),\left( {d,.2} \right)} \right\}$ is a normal fuzzy set and $B = \left\{ {\left( {a,.3} \right),\left( {b,.5} \right),\left( {c,0} \right),\left( {d,.8} \right)} \right\}$ is a subnormal fuzzy set.

Cardinality of a Fuzzy Set

A fuzzy set $A$ defined on a set$X$, where$X$ is finite. The scale cardinality of $A$ is defined as
$\left| A \right| = \sum\limits_{x \in X} {{\mu _A}(x)}$

Example of cardinality of a fuzzy set

Cardinality of the fuzzy set $A = \left\{ {\left( {a,.3} \right),\left( {b,.5} \right),\left( {c,1} \right),\left( {d,.2} \right)} \right\}$ on $X = \left\{ {a,b,c,d} \right\}$ is

$\left| A \right| = .3 + .5 + 1 + .2 = 2.0$

Nucleus of a fuzzy set

The nucleus of a fuzzy set $A$is defined by

$neucleous(A) = \left\{ {x \in X:{\mu _A}(x) = 1} \right\}$ 

If there is only one point such that its membership degree is equal to 1, then this point is called as the peak value of the fuzzy set$A$.

Example of nucleus of a fuzzy set

The nucleus of the fuzzy set $A = \left\{ {\left( {a,.3} \right),\left( {b,.5} \right),\left( {c,1} \right),\left( {d,.2} \right)} \right\}$ on $X = \left\{ {a,b,c,d} \right\}$ is $\left\{ c \right\}$ .

Equality of two Fuzzy sets

Two fuzzy sets are said to be equal, if they have same number of elements and their membership function are also equal. i.e., if $A$ and$B$ are two fuzzy sets on $X$ then$A$ and$B$
are said to be equal and is denoted by if and only if for every

${\mu _A}(x) = {\mu _B}(x)\forall x \in X$ 

Example of inequality of two fuzzy set

$A = \left\{ {\left( {a,.3} \right),\left( {b,.5} \right),\left( {c,1} \right),\left( {d,.2} \right)} \right\}$ and $B = \left\{ {\left( {a,.3} \right),\left( {b,.5} \right),\left( {c,0} \right),\left( {d,.2} \right)} \right\}$ are not equal fuzzy sets, since ${\mu _A}(c) \ne {\mu _B}(c)$ .

Union of Two Fuzzy Sets

Let $A$ and$B$ be two fuzzy sets defined on$X$. Union of $A$ and$B$ is defined by the membership function

${\mu _{A \cup B}}(x) = \max \left\{ {{\mu _A}(x),{\mu _B}(x)} \right\}\forall x \in X$ 

Example of union of two fuzzy sets

Consider two fuzzy sets  $A = \left\{ {\left( {a,.2} \right),\left( {b,.6} \right),\left( {c,.7} \right),\left( {d,.2} \right)} \right\}$ and $B = \left\{ {\left( {a,.3} \right),\left( {b,.4} \right),\left( {c,.9} \right),\left( {d,.1} \right)} \right\}$ on $X = \left\{ {a,b,c,d} \right\}$ .

 ${\mu _{A \cup B}}(a) = \max \left\{ {{\mu _A}(a),{\mu _B}(a)} \right\} = \max \left\{ {.2,.3} \right\} = .3$

${\mu _{A \cup B}}(b) = \max \left\{ {{\mu _A}(b),{\mu _B}(b)} \right\} = \max \left\{ {.6,.4} \right\} = .6$

${\mu _{A \cup B}}(c) = \max \left\{ {{\mu _A}(c),{\mu _B}(c)} \right\} = \max \left\{ {.7,.9} \right\} = .9$

${\mu _{A \cup B}}(d) = \max \left\{ {{\mu _A}(d),{\mu _B}(d)} \right\} = \max \left\{ {.2,.1} \right\} = .2$

$\therefore A \cup B = \left\{ {\left( {a,.3} \right),\left( {b,.6} \right),\left( {c,.9} \right),\left( {d,.2} \right)} \right\}$

Intersection of Two Fuzzy Sets

Let $A$ and$B$ be two fuzzy sets defined on$X$. Intersection of $A$ and$B$ is defined by

${\mu _{A \cap B}}(x) = \max \left\{ {{\mu _A}(x),{\mu _B}(x)} \right\}\forall x \in X$ 

Example of intersection of two fuzzy sets

Consider two fuzzy sets  $A = \left\{ {\left( {a,.2} \right),\left( {b,.6} \right),\left( {c,.7} \right),\left( {d,.2} \right)} \right\}$ and $B = \left\{ {\left( {a,.3} \right),\left( {b,.4} \right),\left( {c,.9} \right),\left( {d,.1} \right)} \right\}$ on $X = \left\{ {a,b,c,d} \right\}$ .

 ${\mu _{A \cap B}}(a) = \min \left\{ {{\mu _A}(a),{\mu _B}(a)} \right\} = \min \left\{ {.2,.3} \right\} = .2$

${\mu _{A \cap B}}(b) = \min \left\{ {{\mu _A}(b),{\mu _B}(b)} \right\} = \min \left\{ {.6,.4} \right\} = .4$

${\mu _{A \cap B}}(c) = \min \left\{ {{\mu _A}(c),{\mu _B}(c)} \right\} = \min \left\{ {.7,.9} \right\} = .7$

${\mu _{A \cap B}}(d) = \min \left\{ {{\mu _A}(d),{\mu _B}(d)} \right\} = \min \left\{ {.2,.1} \right\} = .1$

$\therefore A \cap B = \left\{ {\left( {a,.2} \right),\left( {b,.4} \right),\left( {c,.7} \right),\left( {d,.1} \right)} \right\}$

Complement of a Fuzzy Set

The complement of a fuzzy set $A$ is defined by the membership function

${\mu _{\overline A }}(x) = 1 - {\mu _A}(x)\forall x \in X$

Example of complement of fuzzy sets

Consider a fuzzy set  $A = \left\{ {\left( {a,.2} \right),\left( {b,.6} \right),\left( {c,.7} \right),\left( {d,.2} \right)} \right\}$ and on $X = \left\{ {a,b,c,d} \right\}$ .

 ${\mu _{\overline A }}(a) = 1 - {\mu _A}(a) = 1 - .2 = .8$

${\mu _{\overline A }}(b) = 1 - {\mu _A}(b) = 1 - .6 = .4$

${\mu _{\overline A }}(c) = 1 - {\mu _A}(c) = 1 - .7 = .3$

${\mu _{\overline A }}(d) = 1 - {\mu _A}(d) = 1 - .2 = .8$

$\therefore {A^c} = \left\{ {\left( {a,.8} \right),\left( {b,.4} \right),\left( {c,.3} \right),\left( {d,.8} \right)} \right\}$ 

So, we hope in this article you have learned some basics of fuzzy set theory.
The basic terminologies and properties related to fuzzy sets like, alpha-cut, strong alpha-cut, support of a fuzzy set, core of a fuzzy set, normal fuzzy set, subnormal fuzzy set, cardinality, nucleus of a fuzzy set etc. are defined and illustrated with very simple examples for better understanding.
In addition union of two fuzzy sets, intersection of two fuzzy sets and complement of a fuzzy set has been explained with examples for each case.