Tutorial: Fuzzy Logic – I

Introduction to Fuzzy Logic

Fuzzy logic is a mathematical framework for representing degrees of truth rather than restricting membership to only 0 or 1.

In classical or crisp set theory, an element either belongs to a set or does not belong to it.

In fuzzy logic, an element can belong to a set with a membership value anywhere between $0$ and $1$ .

For example, if $x$ represents temperature, we might have:

\mu_{\text{Warm}}(x) = 0.7

This means that $x$ has a membership degree of $0.7$ in the fuzzy set Warm.

1. Crisp Sets

In a classical set, the membership function can have only two values:

\mu_A(x) \in \{0, 1\}

where:

\mu_A(x) = \begin{cases} 1, & \text{if } x \in A \\ 0, & \text{if } x \notin A \end{cases}

Thus, membership is binary.

For example, if $A$ represents the set of even numbers:

A = \{2, 4, 6, 8, \ldots\}

then:

\mu_A(4) = 1

while:

\mu_A(5) = 0

2. Fuzzy Sets

A fuzzy set allows partial membership.

The membership function is defined as:

\mu_A(x) : X \rightarrow [0, 1]

Therefore:

0 \leq \mu_A(x) \leq 1

The values have the following interpretation:

Membership	Meaning
$\mu_A(x) = 0$	No membership
$0 < \mu_A(x) < 1$

For example:

\mu_{\text{Warm}}(25) = 0.8

means that a temperature of $25^\circ$ C has a membership degree of $0.8$ in the fuzzy set Warm.

3. Fuzzy Membership Function

A fuzzy set $A$ can be represented using its membership function:

\mu_A(x)

The function maps each element $x$ to a value between $0$ and $1$ :

\mu_A(x) \in [0, 1]

For example:

\mu_A(x) = \begin{cases} 0, & x \leq 10 \\ \dfrac{x - 10}{10}, & 10 < x < 20 \\ 1, & x \geq 20 \end{cases}

This function gradually increases membership from $0$ to $1$ .

4. Fuzzy Union

The union of two fuzzy sets $A$ and $B$ represents the degree to which an element belongs to at least one of the sets.

The standard fuzzy union is defined using the maximum operator:

\mu_{A \cup B}(x) = \max\left( \mu_A(x), \mu_B(x) \right)

For example, suppose:

\mu_A(x) = 0.4

and:

\mu_B(x) = 0.7

Then:

\mu_{A \cup B}(x) = \max(0.4, 0.7) = 0.7

The larger membership value is selected.

5. Fuzzy Intersection

The intersection of two fuzzy sets represents the degree to which an element belongs to both sets.

The standard fuzzy intersection is defined using the minimum operator:

\mu_{A \cap B}(x) = \min\left( \mu_A(x), \mu_B(x) \right)

For example, if:

\mu_A(x) = 0.4

and:

\mu_B(x) = 0.7

then:

\mu_{A \cap B}(x) = \min(0.4, 0.7) = 0.4

The smaller membership value is selected.

6. Fuzzy Complement

The complement of a fuzzy set represents the degree to which an element does not belong to the set.

The standard complement is:

\mu_{\overline{A}}(x) = 1 - \mu_A(x)

For example, if:

\mu_A(x) = 0.8

then:

\mu_{\overline{A}}(x) = 1 - 0.8 = 0.2

Similarly, if:

\mu_A(x) = 0.3

then:

\mu_{\overline{A}}(x) = 0.7

7. Fuzzy Set Example

Suppose two fuzzy sets describe temperature:

$A$ = Warm
$B$ = Hot

For a particular temperature $x$ , suppose:

\mu_A(x) = 0.6

and:

\mu_B(x) = 0.8

Union

\mu_{A \cup B}(x) = \max(0.6, 0.8) = 0.8

Intersection

\mu_{A \cap B}(x) = \min(0.6, 0.8) = 0.6

Complement of $A$

\mu_{\overline{A}}(x) = 1 - 0.6 = 0.4

8. Important Fuzzy Set Formulas

Membership

0 \leq \mu_A(x) \leq 1

Fuzzy Union

\mu_{A \cup B}(x) = \max\left( \mu_A(x), \mu_B(x) \right)

Fuzzy Intersection

\mu_{A \cap B}(x) = \min\left( \mu_A(x), \mu_B(x) \right)

Fuzzy Complement

\mu_{\overline{A}}(x) = 1 - \mu_A(x)

9. Crisp Sets vs Fuzzy Sets

Feature	Crisp Set	Fuzzy Set
Membership values	$0$ or $1$	$[0, 1]$
Partial membership	No	Yes
Boundary	Sharp	Gradual
Membership function	Binary	Continuous or discrete
Example	Even number	Warm temperature

For a crisp set:

\mu_A(x) \in \{0, 1\}

For a fuzzy set:

\mu_A(x) \in [0, 1]

The key difference is that fuzzy logic allows partial membership.

10. Fuzzy Set Representation

A finite fuzzy set can be represented as a collection of elements with their membership values.

For example:

A = \bigl\{ (1, 0.2),\ (2, 0.5),\ (3, 0.8),\ (4, 1.0) \bigr\}

This means:

\mu_A(1) = 0.2,\quad \mu_A(2) = 0.5,\quad \mu_A(3) = 0.8,\quad \mu_A(4) = 1

The membership values describe the degree to which each element belongs to the fuzzy set.

11. Fuzzy Operations Example

Consider:

\mu_A(x) = 0.3

and:

\mu_B(x) = 0.9

Then the union is:

\mu_{A \cup B}(x) = \max(0.3, 0.9) = 0.9

The intersection is:

\mu_{A \cap B}(x) = \min(0.3, 0.9) = 0.3

The complement of $A$ is:

\mu_{\overline{A}}(x) = 1 - 0.3 = 0.7

The complement of $B$ is:

\mu_{\overline{B}}(x) = 1 - 0.9 = 0.1

12. De Morgan's Laws for Fuzzy Sets

The standard fuzzy operations satisfy analogues of De Morgan's laws.

First law:

\overline{A \cup B} = \overline{A} \cap \overline{B}

In terms of membership functions:

\mu_{\overline{A \cup B}}(x) = \min\bigl(1 - \mu_A(x),\ 1 - \mu_B(x)\bigr)

Second law:

\overline{A \cap B} = \overline{A} \cup \overline{B}

Therefore:

\mu_{\overline{A \cap B}}(x) = \max\bigl(1 - \mu_A(x),\ 1 - \mu_B(x)\bigr)

13. Important Properties

Fuzzy union using the maximum operator satisfies:

Commutativity

A \cup B = B \cup A

A \cap B = B \cap A

Associativity

(A \cup B) \cup C = A \cup (B \cup C)

(A \cap B) \cap C = A \cap (B \cap C)

Idempotency

A \cup A = A

A \cap A = A

These properties make fuzzy set operations mathematically consistent with the corresponding standard set operations.

14. Summary

Fuzzy logic extends classical set theory by allowing elements to have degrees of membership.

The membership function is:

\mu_A(x) \in [0, 1]

The three fundamental fuzzy-set operations are:

Union

\mu_{A \cup B}(x) = \max\bigl(\mu_A(x),\ \mu_B(x)\bigr)

Intersection

\mu_{A \cap B}(x) = \min\bigl(\mu_A(x),\ \mu_B(x)\bigr)

Complement

\mu_{\overline{A}}(x) = 1 - \mu_A(x)

The key idea to remember is:

\text{Fuzzy Logic} = \text{Degree of Membership}

Unlike crisp logic, where membership is restricted to $0$ or $1$ , fuzzy logic allows intermediate values such as:

0.2,\quad 0.5,\quad 0.7,\quad 0.9

This makes fuzzy logic useful for representing concepts such as temperature, speed, height, pressure, risk, and intelligence, where boundaries are not always naturally binary.

Next Topics

After fuzzy sets and their basic operations, the natural next topics are:

Fuzzy Relations
Fuzzy Cartesian Product
Fuzzy Composition
Fuzzy Rules
Linguistic Variables
Fuzzy Inference Systems
Mamdani Fuzzy Inference
Sugeno Fuzzy Inference
Fuzzification
Defuzzification
Centroid Method
Applications of Fuzzy Logic

Tutorial: Fuzzy Logic – I

Introduction to Fuzzy Logic

1. Crisp Sets

2. Fuzzy Sets

3. Fuzzy Membership Function

4. Fuzzy Union

5. Fuzzy Intersection

6. Fuzzy Complement

7. Fuzzy Set Example

Union

Intersection

Complement of AAA

8. Important Fuzzy Set Formulas

Membership

Fuzzy Union

Fuzzy Intersection

Fuzzy Complement

9. Crisp Sets vs Fuzzy Sets

10. Fuzzy Set Representation

11. Fuzzy Operations Example

12. De Morgan's Laws for Fuzzy Sets

13. Important Properties

Commutativity

Associativity

Idempotency

14. Summary

Union

Intersection

Complement

Next Topics

Complement of $A$