240 
 

 American Scientific Research Journal for Engineering, Technology,  and Sciences  (ASRJETS) 
ISSN (Print) 2313-4410, ISSN (Online) 2313-4402 

© Global Society of Scientific Research and Researchers  
http://asrjetsjournal.org/  

 

A Comparative Study of Classical Relation and Fuzzy 

Relation 

Abdul Alim* 

Lecturer, Dept. of Natural Sciences (Mathematics), BGMEA University of Fashion & Technology, Dhaka-1230, 

Bangladesh 

Email: abdulalim@buft.edu.bd 

 

 

Abstract 

Fuzzy relation is an extension of classical relation as fuzzy set is an extension of classical set. This work 

represents an overview of classical relation and fuzzy relation. Some definitions related to fuzzy relation are 

discussed and also explained by giving example. Max-min operation and max-product operation are discussed 

for both classical relations and fuzzy relations. Max-avg operation is also discussed for fuzzy relations. Finally, 

fuzzy equivalence relation is discussed. 

Keywords: Characteristic function; Membership grade; Max-min operation; Max-product operation; Max-avg 

operation. 

1. Introduction 

The concept of fuzzy set was first introduced by L. A. Zadeh in 1965. Fuzzy set is an extension or generalization 

of classical set [1]. A classical set is a well defined collection of objects. Each classical set can be defined by a 

characteristics function [5], 

𝜇𝜇𝑋𝑋(𝑥𝑥) = �1 ; 𝑥𝑥 ∈ 𝑋𝑋
0 ; 𝑥𝑥 ∉ 𝑋𝑋 

The characteristics function indicates whether an element is present in the set or not. If any element is present in 

the set 𝑋𝑋, the membership grade is 1 and the membership grade is 0 if it is not present in the set.   

------------------------------------------------------------------------ 

* Corresponding author.  

http://asrjetsjournal.org/


American Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2017) Volume 31, No  1, pp 240-248 

241 
 

If the valuation set {0,1} is replaced by the real interval [0,1], the set is called fuzzy set and various degrees of 

membership are obtained. Thus, a fuzzy set is obviously a generalization of classical set. 

In mathematics, a relation describes how the elements of one set, 𝑋𝑋 are associated with the elements of another 

set, 𝑌𝑌. If 𝑋𝑋 and 𝑌𝑌 be two sets then relation between 𝑋𝑋 and 𝑌𝑌is a subset of the Cartesian product 𝑋𝑋 × 𝑌𝑌.That is, 

the relation, 𝑅𝑅 between 𝑋𝑋 and 𝑌𝑌 can be defined by, 

𝑅𝑅 = {(𝑥𝑥,𝑦𝑦)|𝑥𝑥 ∈ 𝑋𝑋,𝑦𝑦 ∈ 𝑌𝑌} 

The characteristics function of a classical relation is given by, 

𝜇𝜇𝑅𝑅(𝑥𝑥,𝑦𝑦) = �1 ; (𝑥𝑥,𝑦𝑦) ∈ 𝑅𝑅
0 ; (𝑥𝑥,𝑦𝑦) ∉ 𝑅𝑅 

Fuzzy relation is the generalization of classical relation as fuzzy set is the generalization of classical set [4]. In a 

fuzzy relation, membership grade 𝑅𝑅(𝑥𝑥,𝑦𝑦) is taken from the interval [0,1] instead of {0, 1}. 

 

2. Fuzzy Relation [4] 

First we describe a classical relation between the color of a mango, 𝑋𝑋 and the grade of its maturity, 𝑌𝑌. 

𝑋𝑋 = {𝑦𝑦𝑦𝑦𝑦𝑦𝑦𝑦𝑦𝑦𝑦𝑦, 𝑦𝑦𝑙𝑙𝑙𝑙ℎ𝑡𝑡 𝑙𝑙𝑔𝑔𝑦𝑦𝑦𝑦𝑔𝑔,𝑙𝑙𝑔𝑔𝑦𝑦𝑦𝑦𝑔𝑔 } 

𝑌𝑌 = {𝑔𝑔𝑟𝑟𝑦𝑦, ℎ𝑟𝑟𝑦𝑦𝑎𝑎 − 𝑚𝑚𝑟𝑟𝑡𝑡𝑚𝑚𝑔𝑔𝑦𝑦𝑚𝑚, 𝑔𝑔𝑙𝑙𝑟𝑟𝑦𝑦} 

Now, the membership grade of classical relation from 𝑋𝑋 to 𝑌𝑌 is given in the tabular form, 

Table 1 

 Raw Half-matured Ripe 

Yellow 0 0 1 

Light green 0 1 0 

Green 1 0 0 

 

We see that, the membership grade is obtained from the set, {0,1}. If the color of the mango is green then it is 

raw, if the color is light green then it is half-matured and if the color is yellow then the mango is ripe. Thus the 

characteristics function of a classical relation is given by, 

𝜇𝜇𝑅𝑅(𝑥𝑥,𝑦𝑦) = �1 ; (𝑥𝑥,𝑦𝑦) ∈ 𝑅𝑅
0 ; (𝑥𝑥,𝑦𝑦) ∉ 𝑅𝑅 



American Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2017) Volume 31, No  1, pp 240-248 

242 
 

Now, if we take the membership grade from [0,1] interval instead of {0,1}, we get a membership grade in fuzzy 

relation. We show the membership grade in the fuzzy relation as follows: 

Table 2 

 Raw Half-matured Ripe 

Yellow 0 0.8 1 

Light Green 0.6 1 0.3 

Green 1 0.3 0 

 

Definition: Let, 𝑋𝑋 and 𝑌𝑌 are two non-empty sets. Then the fuzzy relations, 𝑅𝑅:𝑋𝑋 × 𝑌𝑌 → 𝐼𝐼 is defined as follows:  

𝑅𝑅(𝑥𝑥,𝑦𝑦) = {(𝑥𝑥,𝑦𝑦), 𝜇𝜇𝑅𝑅(𝑥𝑥,𝑦𝑦)|(𝑥𝑥,𝑦𝑦) ∈ 𝑋𝑋 × 𝑌𝑌} 

Where, 𝜇𝜇𝑅𝑅(𝑥𝑥,𝑦𝑦)is the membership grade of the ordered pair (𝑥𝑥,𝑦𝑦)and  𝐼𝐼 = [0,1] [3]. 

Let, 𝑋𝑋 = {𝑥𝑥1, 𝑥𝑥2, … … , 𝑥𝑥𝑚𝑚} and  𝑌𝑌 = {𝑦𝑦1,𝑦𝑦2, … … ,𝑦𝑦𝑛𝑛} , the 𝑚𝑚 × 𝑔𝑔 matrix form of the fuzzy relation is: 

𝑅𝑅(𝑥𝑥,𝑦𝑦) = �
𝜇𝜇𝑅𝑅(𝑥𝑥1,𝑦𝑦1) ⋯ 𝜇𝜇𝑅𝑅(𝑥𝑥1,𝑦𝑦𝑛𝑛)

⋮ ⋱ ⋮
𝜇𝜇𝑅𝑅(𝑥𝑥𝑚𝑚,𝑦𝑦1) ⋯ 𝜇𝜇𝑅𝑅(𝑥𝑥𝑚𝑚 ,𝑦𝑦𝑛𝑛)

� 

Example: Let, 𝑋𝑋 = {0,1,2,3,4,5} and 𝑌𝑌 = {1,3,5,7,9,11}. A fuzzy relation is defined as follows: 

𝑅𝑅(𝑥𝑥,𝑦𝑦) =

⎩
⎨

⎧
1  ;   𝑥𝑥 ≤ 0

1 −
𝑥𝑥2

1 + 𝑦𝑦2
 ; 0 < 𝑥𝑥 ≤ 𝑦𝑦

0   ;    𝑥𝑥 > 𝑦𝑦

 

The membership grade of the relation is given in the following table: 

Table 3 

𝑹𝑹(𝒙𝒙,𝒚𝒚) 1 3 5 7 9 11 

0 1 1 1 1 1 1 

1 0.5 0.9 0.96 0.98 0.988 0.99 

2 0 0.6 0.85 0.92 0.95 0.97 

3 0 0.1 0.65 0.82 0.89 0.93 

4 0 0 0.38 0.68 0.80 0.87 

5 0 0 0.04 0.5 0.7 0.8 

 



American Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2017) Volume 31, No  1, pp 240-248 

243 
 

2.1. Domain, Range and Height of a Fuzzy Relation [2,5]  

Let, 𝑅𝑅:𝑋𝑋 × 𝑌𝑌 → 𝐼𝐼 be a fuzzy relation. Then domain of 𝑅𝑅(𝑥𝑥,𝑦𝑦) is defined as follows: 

𝐷𝐷𝑦𝑦𝑚𝑚(𝑅𝑅) = ��max
𝑦𝑦

𝑅𝑅(𝑥𝑥,𝑦𝑦)� �(𝑥𝑥,𝑦𝑦) ∈ 𝑋𝑋 × 𝑌𝑌� 

This is also called 1st projection or projection of 𝑅𝑅(𝑥𝑥,𝑦𝑦) on 𝑋𝑋. 

 The range of  𝑅𝑅(𝑥𝑥,𝑦𝑦) is defined as follows: 

𝑅𝑅𝑟𝑟𝑔𝑔𝑙𝑙𝑦𝑦(𝑅𝑅) = ��max
𝑥𝑥

𝑅𝑅(𝑥𝑥,𝑦𝑦)� �(𝑥𝑥,𝑦𝑦) ∈ 𝑋𝑋 × 𝑌𝑌� 

This is also called 2nd projection or projection of 𝑅𝑅(𝑥𝑥,𝑦𝑦) on 𝑌𝑌. 

And the height of 𝑅𝑅(𝑥𝑥,𝑦𝑦) is defined as follows: 

𝐻𝐻(𝑅𝑅) = max
𝑦𝑦

�max
𝑥𝑥

𝑅𝑅(𝑥𝑥,𝑦𝑦)� |(𝑥𝑥,𝑦𝑦) ∈ 𝑋𝑋 × 𝑌𝑌 

This is also called global projection. 

Note: If the height of a fuzzy relation, 𝐻𝐻(𝑅𝑅) = 1,then the fuzzy relation is called normal fuzzy relation and if, 

𝐻𝐻(𝑅𝑅) < 1, the fuzzy relation is called subnormal fuzzy relation [2]. 

Example: Let, 𝑋𝑋 = {1,2,3,4,5} and 𝑌𝑌 = {3,4,5,6,7,8}. A fuzzy relation is defined as follows: 

𝑅𝑅(𝑥𝑥,𝑦𝑦) = �1 −
𝑥𝑥2

2𝑦𝑦2
  ; 0 < 𝑥𝑥 ≤ 𝑦𝑦

0       ;  𝑥𝑥 > 𝑦𝑦
 

The membership grade of the relation is given in the following table: 

Table 4 

𝑹𝑹(𝒙𝒙,𝒚𝒚) 3 4 5 6 7 8 

1 0.94 0.97 0.98 0.986 0.989 0.99 

2 0.78 0.875 0.92 0.94 0.96 0.97 

3 0.5 0.72 0.82 0.875 0.91 0.93 

4 0 0.5 0.68 0.78 0.84 0.875 

5 0 0 0.5 0.65 0.74 0.80 

 



American Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2017) Volume 31, No  1, pp 240-248 

244 
 

Thus we get, 

𝐷𝐷𝑦𝑦𝑚𝑚(𝑅𝑅) = {0.99, 0.97, 0.93, 0.875, 0.80} 

𝑅𝑅𝑟𝑟𝑔𝑔𝑙𝑙𝑦𝑦(𝑅𝑅) = {0.94, 0.97, 0.986, 0.989, 0.99} 

And,  

𝐻𝐻(𝑅𝑅) = 0.99 

3. Standard Operations on relations 

We will discuss three standard operations on fuzzy relations. These are max-min operation, max-product 

operation and max-avg operation. 

3.1. Max-Min Operation [2] 

Definition:  If 𝑃𝑃 and 𝑄𝑄 are two fuzzy relations, then max-Min operation of 𝑃𝑃 and 𝑄𝑄 is defined by, 

𝑃𝑃𝑦𝑦𝑄𝑄 = {𝑚𝑚𝑙𝑙𝑔𝑔(𝑃𝑃(𝑥𝑥,𝑦𝑦),𝑄𝑄(𝑦𝑦, 𝑧𝑧)};∀(𝑥𝑥,𝑦𝑦), (𝑦𝑦, 𝑧𝑧) ∈ 𝑋𝑋 × 𝑌𝑌𝑦𝑦∈𝑌𝑌
𝑚𝑚𝑚𝑚𝑥𝑥  

Example: If 𝑃𝑃 = �
1 0 1
1 1 1
0 1 1

� and 𝑄𝑄 = �
1 1 0
1 0 1
0 0 1

    
1
0
1
� be two classical relations, then max-min operations on 𝑃𝑃 

and 𝑄𝑄 is: 

𝑃𝑃𝑦𝑦𝑄𝑄 = �
∨ (1 ∧ 1,0 ∧ 1,1 ∧ 0) ∨ (1 ∧ 1,0 ∧ 0,1 ∧ 0) ∨ (1 ∧ 0,0 ∧ 1,1 ∧ 1)
∨ (1 ∧ 1,1 ∧ 1,1 ∧ 0) ∨ (1 ∧ 1,1 ∧ 0,1 ∧ 0) ∨ (1 ∧ 0,1 ∧ 1,1 ∧ 1)
∨ (0 ∧ 1,1 ∧ 1,1 ∧ 0) ∨ (0 ∧ 1,1 ∧ 0,1 ∧ 0) ∨ (0 ∧ 0,1 ∧ 1,1 ∧ 1)

    
∨ (1 ∧ 1,0 ∧ 0,1 ∧ 1)
∨ (1 ∧ 1,1 ∧ 0,1 ∧ 1)
∨ (0 ∧ 1,1 ∧ 0,1 ∧ 1)

� 

= �
∨ (1,0,0) ∨ (1,0,0) ∨ (0,0,1)
∨ (1,1,0) ∨ (1,0,0) ∨ (0,1,1)
∨ (0,1,0) ∨ (0,0,0) ∨ (0,1,1)

    
∨ (1,0,1)
∨ (1,0,1)
∨ (0,0,1)

� 

= �
1 1 1
1 1 1
1 0 1

    
1
1
1
� 

Example: If 𝑃𝑃 = �
. 2 . 4 . 7
0 . 6 1
. 5 . 7 . 5

� and 𝑄𝑄 = �
1 . 6 . 8
. 3 . 2 0
0 1 . 5

    
. 7
. 9
. 5
� be two fuzzy relations, then max-min operations on 

𝑃𝑃 and 𝑄𝑄 is: 

𝑃𝑃𝑦𝑦𝑄𝑄

= �
∨ (.2 ∧ 1, .4 ∧ .3, .7 ∧ 0) ∨ (.2 ∧ .6, .4 ∧ .2, .7 ∧ 1) ∨ (.2 ∧ .8, .4 ∧ 0, .7 ∧ .5)
∨ (0 ∧ 1, .6 ∧ .3,1 ∧ 0) ∨ (0 ∧ .6, .6 ∧ .2,1 ∧ 1) ∨ (0 ∧ .8, .6 ∧ 0,1 ∧ .5)
∨ (.5 ∧ 1, .7 ∧ .3, .5 ∧ 0) ∨ (.5 ∧ .6, .7 ∧ .2, .5 ∧ 1) ∨ (.5 ∧ .8, .7 ∧ 0, .5 ∧ .5)

    
∨ (.2 ∧ .7, .4 ∧ .9, .7 ∧ .5)
∨ (0 ∧ .7, .6 ∧ .9,1 ∧ .5)
∨ (.5 ∧ .7, .7 ∧ .9, .5 ∧ .5)

� 



American Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2017) Volume 31, No  1, pp 240-248 

245 
 

= �
∨ (.2, .3,0) ∨ (.2, .2, .7) ∨ (.2,0, .5)
∨ (0, .3,0) ∨ (0, .2,1) ∨ (0,0, .5)
∨ (.5, .3,0) ∨ (.5, .2, .5) ∨ (.5,0, .5)

    
∨ (.2, .4, .5)
∨ (0, .6, .5)
∨ (.5, .7, .5)

� 

= �
. 3 . 7 . 5
. 3 1 . 5
. 5 . 5 . 5

    
. 5
. 6
. 7
� 

3.2. Max-Product Operation [4]  

Definition:  If 𝑃𝑃 and 𝑄𝑄 are two fuzzy relations, then max-product operation of 𝑃𝑃 and 𝑄𝑄 is defined by, 

𝑃𝑃𝑦𝑦𝑄𝑄 = {(𝑃𝑃(𝑥𝑥,𝑦𝑦).𝑄𝑄(𝑦𝑦, 𝑧𝑧)};∀(𝑥𝑥,𝑦𝑦), (𝑦𝑦, 𝑧𝑧) ∈ 𝑋𝑋 × 𝑌𝑌𝑦𝑦∈𝑌𝑌
𝑚𝑚𝑚𝑚𝑥𝑥  

Example: If 𝑃𝑃 = �
1 0 1
1 1 1
0 1 1

� and 𝑄𝑄 = �
1 1 0
1 0 1
0 0 1

    
1
0
1
� be two classical relations, then max-product operations on 

𝑃𝑃 and 𝑄𝑄 is: 

𝑃𝑃𝑦𝑦𝑄𝑄 = �
∨ (1.1, 0.1, 1.0) ∨ (1.1, 0.0, 1.0) ∨ (1.0, 0.1, 1.1)
∨ (1.1, 1.1, 1.0) ∨ (1.1, 1.0, 1.0) ∨ (1.0, 1.1, 1.1)
∨ (0.1, 1.1, 1.0) ∨ (0.1, 1.0, 1.0) ∨ (0.0, 1.1, 1.1)

    
∨ (1.1, 0.0, 1.1)
∨ (1.1, 1.0, 1.1)
∨ (0.1, 1.0, 1.1)

� 

= �
∨ (1,0,0) ∨ (1,0,0) ∨ (0,0,1)
∨ (1,1,0) ∨ (1,0,0) ∨ (0,1,1)
∨ (0,1,0) ∨ (0,0,0) ∨ (0,1,1)

    
∨ (1,0,1)
∨ (1,0,1)
∨ (0,0,1)

� 

= �
1 1 1
1 1 1
1 0 1

    
1
1
1
� 

Example: If 𝑃𝑃 = �
. 2 . 4 . 7
0 . 6 1
. 5 . 7 . 5

�  and 𝑄𝑄 = �
1 . 6 . 8
. 3 . 2 0
0 1 . 5

    
. 7
. 9
. 5
�  be two fuzzy relations, then max-product 

operations on 𝑃𝑃 and 𝑄𝑄 is: 

𝑃𝑃𝑦𝑦𝑄𝑄 = �
∨ (.2, .12,0) ∨ (.12, .08, .7) ∨ (.16,0, .35)
∨ (0, .18,0) ∨ (0, .12,1) ∨ (0,0, .5)
∨ (.5, .21,0) ∨ (.3, .14, .5) ∨ (.4,0, .25)

    
∨ (.14, . .36, .35)
∨ (0, .54, .5)

∨ (.35, .63, .25)
� 

= �
. 2 . 7 . 35

. 18 1 . 5
. 5 . 5 . 4

    
. 36
. 54
. 63

� 

3.3. Max-Avg Operation [4] 

Definition:  If 𝑃𝑃 and 𝑄𝑄 are two fuzzy relations, then max-avg operation of 𝑃𝑃 and 𝑄𝑄 is defined by, 



American Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2017) Volume 31, No  1, pp 240-248 

246 
 

𝑃𝑃𝑦𝑦𝑄𝑄 = {
1
2

(𝑃𝑃(𝑥𝑥,𝑦𝑦) + 𝑄𝑄(𝑦𝑦, 𝑧𝑧)};∀(𝑥𝑥,𝑦𝑦), (𝑦𝑦, 𝑧𝑧) ∈ 𝑋𝑋 × 𝑌𝑌𝑦𝑦∈𝑌𝑌
𝑚𝑚𝑚𝑚𝑥𝑥  

Example: If 𝑃𝑃 = �
. 2 . 4 . 7
0 . 6 1
. 5 . 7 . 5

� and 𝑄𝑄 = �
1 . 6 . 8
. 3 . 2 0
0 1 . 5

    
. 7
. 9
. 5
� be two fuzzy relations, then max-avg operations on 

𝑃𝑃 and 𝑄𝑄 is: 

𝑃𝑃𝑦𝑦𝑄𝑄

=

⎣
⎢
⎢
⎢
⎢
⎡∨ (

. 2 + 1
2

,
. 4 + .3

2
,
. 7 + 0

2
) ∨ (

. 2 + .6
2

,
. 4 + .2

2
,
. 7 + 1

2
) ∨ (

. 2 + .8
2

,
. 4 + 0

2
,
. 7 + .5

2
)

∨ (
0 + 1

2
,
. 6 + .3

2
,
1 + 0

2
) ∨ (

0 + .6
2

,
. 6 + .2

2
,
1 + 1

2
) ∨ (

0 + .8
2

,
. 6 + 0

2
,
1 + .5

2
)

∨ (
. 5 + 1

2
,
. 7 + .3

2
,
. 5 + 0

2
) ∨ (

. 5 + .6
2

 ,
. 7 + .2

2
,
. 5 + 1

2
) ∨ (

. 5 + .8
2

,
. 7 + 0

2
,
. 5 + .5

2
)

    

∨ (
. 2 + .7

2
,
. 4 + .9

2
,
. 7 + .5

2
)

∨ (
0 + .7

2
,
. 6 + .9

2
,
1 + .5

2
)

∨ (
. 5 + .7

2
,
. 7 + .9

2
,
. 5 + .5

2
)⎦
⎥
⎥
⎥
⎥
⎤

 

= �
∨ (.6, .35, .35) ∨ (. .4, .3, . .85) ∨ (.5, .2, .6)
∨ (.5, .45, .5) ∨ (.3, .4,1) ∨ (.4, .3, .75)
∨ (.75, .5, .25) ∨ (.55, .45, .75) ∨ (.65, .35, .5)

    
∨ (.45, .65, .6)
∨ (.35, .75, .75)
∨ (.6, .8, .5)

� 

= �
. 6 . 85 . 6
. 5 1 . 75

. 75 . 75 . 65
    

. 65

. 75
. 8
� 

From the above example we see that, max-min operation and max-product operation on two classical relations 

give the same result. But max-min operation and max-product operation on two fuzzy relations have different 

result. Also the max-avg operation on two fuzzy relations has different result from max-min and max-product 

operation. 

4. Fuzzy Equivalence relation [2] 

A binary fuzzy relation 𝑅𝑅  in 𝑋𝑋 × 𝑋𝑋  is called fuzzy equivalence relation if it satisfies the following three 

conditions: 

i) 𝑅𝑅(𝑥𝑥, 𝑥𝑥) = 1 ;∀𝑥𝑥 ∈ 𝑋𝑋      [Reflexive condition] 

ii) 𝑅𝑅(𝑥𝑥,𝑦𝑦) = 𝑅𝑅(𝑦𝑦, 𝑥𝑥) ;  ∀𝑥𝑥,𝑦𝑦 ∈ 𝑋𝑋     [Symmetric condition] 

iii) 𝑅𝑅(𝑥𝑥, 𝑧𝑧) ≥ {𝑚𝑚𝑙𝑙𝑔𝑔(𝑅𝑅(𝑥𝑥,𝑦𝑦),𝑅𝑅(𝑦𝑦, 𝑧𝑧)} ;∀𝑥𝑥,𝑦𝑦, 𝑧𝑧 ∈ 𝑋𝑋𝑦𝑦∈𝑌𝑌
𝑚𝑚𝑚𝑚𝑥𝑥  [Transitive condition] 

Example: Suppose,   

              𝑥𝑥1 𝑥𝑥2 𝑥𝑥3 

𝑅𝑅 =
𝑥𝑥1
𝑥𝑥2
𝑥𝑥3
�
1 0 0
0 1 . 6
0 . 6 1

� 

For reflexivity, 



American Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2017) Volume 31, No  1, pp 240-248 

247 
 

𝑅𝑅(𝑥𝑥1, 𝑥𝑥1) = 1 

𝑅𝑅(𝑥𝑥2, 𝑥𝑥2) = 1 

𝑅𝑅(𝑥𝑥3, 𝑥𝑥3) = 1 

Therefore, 𝑅𝑅 is reflexive.                   

For symmetry, 

𝑅𝑅(𝑥𝑥1, 𝑥𝑥2) = 𝑅𝑅(𝑥𝑥2, 𝑥𝑥1) = 0 

𝑅𝑅(𝑥𝑥1, 𝑥𝑥3) = 𝑅𝑅(𝑥𝑥3, 𝑥𝑥1) = 0 

𝑅𝑅(𝑥𝑥2, 𝑥𝑥3) = 𝑅𝑅(𝑥𝑥3, 𝑥𝑥2) = .6 

        Therefore,  𝑅𝑅 is symmetric.                                                                     

 For transitivity, 

𝑅𝑅 = �
 1 0 0
 0 1 . 6
 0 . 6 1

 � 

Now,  

𝑅𝑅𝑦𝑦𝑅𝑅 = �
 1 0 0
 0 1 . 6
 0 . 6 1

 � o �
 1 0 0
 0 1 . 6
 0 . 6 1

 � 

= �
 ∨ (1 ∧ 1, 0 ∧ 0, 0 ∧ 0)  ∨ (1 ∧ 0, 0 ∧ 1, 0 ∧ .6)  ∨ (1 ∧ 0, 0 ∧ 0, 0 ∧ 0)
∨ (0 ∧ 1, 1 ∧ 0, .6 ∧ 0) ∨ (0 ∧ 0, 1 ∧ 1, .6 ∧ .6) ∨ (0 ∧ 0, 1 ∧ .6, .6 ∧ 1)
 ∨ (0 ∧ 1, .6 ∧ 0, 1 ∧ 0)  ∨ (0 ∧ 0, .6 ∧ 1, 1 ∧ .6)  ∨ (0 ∧ 0, .6 ∧ .6, 1 ∧ 1)

� 

= �
 ∨ (1, 0, 0) ∨ (0, 0, 0)  ∨ (0, 0, 0)
∨ (0, 0, 0) ∨ (0, 1, .6) ∨ (0, .6, .6)
  ∨ (0, 0, 0) ∨ (0, .6, .6)  ∨ (0, .6, 1)

� 

= �
1 0 0
 0 1 . 6
 0 . 6 1

� = R 

That is, 𝑅𝑅𝑦𝑦𝑅𝑅 ≥ 𝑅𝑅. So, 𝑅𝑅 is transitive. 

Since, the relation 𝑅𝑅 satisfies all the three conditions, it is an equivalence relation. 



American Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2017) Volume 31, No  1, pp 240-248 

248 
 

Example: Consider another fuzzy relation, 𝑅𝑅 = �
1 0 . 6
0 1 . 3
. 6 . 3 1

�. The relation is not equivalence relation, because 

it doesn’t satisfy the condition of transitivity.  

5. Conclusion 

Fuzzy relations generalize the concept of classical relations in the same manner as fuzzy sets generalize the 

concept of classical sets. Fuzzy set theory as well as fuzzy relations has significant importance in mathematics 

and is being used widely in many areas such as clustering, decision-making, medical diagnosis, engineering 

sector, stock market, even in the cricket field to predict score. 

References 

[1] L. A. Zadeh, , Fuzzy Sets, Information and Control, 8(1965) 338-353. 

[2] George J, Klir, Bo Yuan, Fuzzy Sets and Fuzzy Logic, Theory and Applications, Prentice Hall Inc. N.J. 

U.S.A. 1995. 

[3] H. J. Zimmermann, Fuzzy Set Theory and its Applications, Third Edition, Kluwer Academic Publishers, 

Boston, U.S.A. 1995 

[4] Majid Hussain, Fuzzy Relations, Thesis for the degree Master of Science (two years) in Mathematical 

Modeling and Simulation, May 2010. 

[5] D. Dubis, H. Prade, Fuzzy Sets and Systems, Theory and Applications, New York, 1980.