Problem 70 - Discrete Mathematics with Applications

Question

The sum of two fuzzy sets \(A\) and \(B\) is the fuzzy set \(A \oplus B,\) where \(d_{A \oplus B}(x)=\) 1\(\wedge\left|d_{A}(x)+d_{B}(x)\right|\) itheir difference is the fuzzy set \(A-B,\) where \(d_{A-B}(x)=\) \(0 \vee\left[d_{A}(x)-d_{B}(x)\right] ;\) and their eartesian produet is the fuzzy set \(A \times B\) where \(d_{A \times B}(x, y)=d_{A}(x) \wedge d_{B}(x) .\) Use the fuzzy sets \(A=\\{\text { Angelo } 0.4, \text { Bart }\) \(0.7,\) Cathy 0.6\(\\}\) and \(B=\\{\operatorname{Dan} 0.3, \text { Elsie } 0.8, \text { Frank } 0.4\\}\) to find each fuzzy set. $$ A^{\prime} $$

Step-by-Step Solution

Answer

The sum of fuzzy sets A and B is: \( A \oplus B = \{ \text{Angelo} : 0.7, \text{Bart} : 1, \text{Cathy} : 1 \} \) The difference of fuzzy sets A and B is: \( A - B = \{ \text{Angelo} : 0.1, \text{Bart} : 0, \text{Cathy} : 0.2 \} \) The cartesian product of fuzzy sets A and B is: \( A \times B = \{ (\text{Angelo}, \text{Dan}) : 0.3, (\text{Angelo}, \text{Elsie}) : 0.4, (\text{Angelo}, \text{Frank}) : 0.4, (\text{Bart}, \text{Dan}) : 0.3, (\text{Bart}, \text{Elsie}) : 0.7, (\text{Bart}, \text{Frank}) : 0.4, (\text{Cathy}, \text{Dan}) : 0.3, (\text{Cathy}, \text{Elsie}) : 0.6, (\text{Cathy}, \text{Frank}) : 0.4 \} \)

Step 1: Fuzzy Set A

\( A = \{ \text{Angelo} : 0.4, \text{Bart} : 0.7, \text{Cathy} : 0.6 \} \)

Step 2: Fuzzy Set B

\( B = \{ \text{Dan} : 0.3, \text{Elsie} : 0.8, \text{Frank} : 0.4 \} \) #Step 2: Find The Sum of Fuzzy Sets A and B# Now, we will find the sum of the two fuzzy sets A and B using the given definition. \( d_{A \oplus B}(x) = 1 \wedge \left| d_{A}(x) + da_{B}(x) \right| \)

Step 3: The Sum of A and B

\( A \oplus B = \{ \text{Angelo} : 1 \wedge \left| 0.4 + 0.3 \right|, \text{Bart} : 1 \wedge \left| 0.7 + 0.8 \right|, \text{Cathy} : 1 \wedge \left| 0.6 + 0.4 \right| \} \) \( A \oplus B = \{ \text{Angelo} : 0.7, \text{Bart} : 1, \text{Cathy} : 1 \} \) #Step 3: Find The Difference of Fuzzy Sets A and B# Next, we will find the difference of the two fuzzy sets A and B using the given definition. \( d_{A - B}(x) = 0 \vee \left[ d_{A}(x) - d_{B}(x) \right] \)

Step 4: The Difference of A and B

\( A - B = \{ \text{Angelo} : 0 \vee \left[ 0.4 - 0.3 \right], \text{Bart} : 0 \vee \left[ 0.7 - 0.8 \right], \text{Cathy} : 0 \vee \left[ 0.6 - 0.4 \right] \} \) \( A - B = \{ \text{Angelo} : 0.1, \text{Bart} : 0, \text{Cathy} : 0.2 \} \) #Step 4: Find The Cartesian Product of Fuzzy Sets A and B# Finally, we will find the cartesian product of the two fuzzy sets A and B using the given definition. \( d_{A \times B}(x,y) = d_{A}(x) \wedge d_{B}(x) \)

Step 5: The Cartesian Product of A and B

\( A \times B = \{ (\text{Angelo}, \text{Dan}) : 0.4 \wedge 0.3, (\text{Angelo}, \text{Elsie}) : 0.4 \wedge 0.8, (\text{Angelo}, \text{Frank}) : 0.4 \wedge 0.4, (\text{Bart}, \text{Dan}) : 0.7 \wedge 0.3, (\text{Bart}, \text{Elsie}) : 0.7 \wedge 0.8, (\text{Bart}, \text{Frank}) : 0.7 \wedge 0.4, (\text{Cathy}, \text{Dan}) : 0.6 \wedge 0.3, (\text{Cathy}, \text{Elsie}) : 0.6 \wedge 0.8, (\text{Cathy}, \text{Frank}) : 0.6 \wedge 0.4 \} \) \( A \times B = \{ (\text{Angelo}, \text{Dan}) : 0.3, (\text{Angelo}, \text{Elsie}) : 0.4, (\text{Angelo}, \text{Frank}) : 0.4, (\text{Bart}, \text{Dan}) : 0.3, (\text{Bart}, \text{Elsie}) : 0.7, (\text{Bart}, \text{Frank}) : 0.4, (\text{Cathy}, \text{Dan}) : 0.3, (\text{Cathy}, \text{Elsie}) : 0.6, (\text{Cathy}, \text{Frank}) : 0.4 \} \)

Key Concepts

Fuzzy Set Operations

Fuzzy sets are an extension of traditional sets, where elements can have degrees of membership. These degrees range between 0 and 1, indicating the intensity of membership. Understanding operations on fuzzy sets allows us to handle data that is not precise or uncertain.

Fuzzy set operations include fundamental processes such as:

• Union: Combines elements from two sets. The membership degree is determined by the maximum value from the two sets for each element.
• Intersection: Considers the minimum value of the degrees of membership of the elements common to both sets.
• Complement: Reflects the negation of each element's membership degree in the set.

These operations enable us to manipulate and analyze fuzzy sets, making them crucial in fields like decision making, pattern recognition, and control systems. In our discussion, we'll focus on specific operations such as the sum, difference, and Cartesian product of fuzzy sets.

Sum of Fuzzy Sets

The sum of two fuzzy sets is an operation that combines them while considering their degrees of membership. For fuzzy sets \(A\) and \(B\), the resulting fuzzy set \(A \oplus B\) incorporates the sum of membership degrees. However, the addition is capped at 1. This means for each element, the membership degree in the resulting set doesn’t exceed 1.

The calculation follows:

• Determine the sum of membership values for each corresponding element.
• Use the min (\(\wedge\)) operator with 1 to ensure values are within the valid range.

For example, where \(d_{A}(x) = 0.4\) and \(d_{B}(x) = 0.3\), the summed membership will be \(d_{A \oplus B}(x) = 0.4 + 0.3 = 0.7\), as 0.7 is less than 1, it remains 0.7 in \(A \oplus B\). This process helps keep fuzzy set operations practical, recognizing that combining entities shouldn't escape logical boundaries.

Difference of Fuzzy Sets

The difference operation on fuzzy sets involves subtracting the membership values of one set from the other. Given fuzzy sets \(A\) and \(B\), the fuzzy set \(A - B\) is determined by subtracting the membership value of \(B\) from \(A\).

Here's how it's calculated:

• Subtract the membership degree of each element in \(B\) from the corresponding element in \(A\).
• Apply the max (\(\vee\)) operator with 0 to ensure no negative values arise.

For example, if \(d_{A}(x) = 0.4\) and \(d_{B}(x) = 0.3\), the resulting membership is \(d_{A - B}(x) = 0.4 - 0.3 = 0.1\). Conversely, negative differences result in 0, as seen where \(d_{A}(x) = 0.7\) and \(d_{B}(x) = 0.8\), yielding \(0 \vee [0.7 - 0.8] = 0\). This operation retains meaningful differences within the logical scope of fuzzy computations.

Cartesian Product of Fuzzy Sets

The Cartesian product of fuzzy sets involves pairing elements from two sets to form ordered pairs, with each pair having an associated degree of membership derived from the original sets. For fuzzy sets \(A\) and \(B\), the Cartesian product \(A \times B\) is computed by combining each element of \(A\) with each element of \(B\).

Here's the step-by-step:

• Create pairs of elements using each from \(A\) with every element from \(B\).
• Determine the membership of each pair by taking the min (\(\wedge\)) of the membership degrees from both sets.

For instance, the pair \((\text{Angelo}, \text{Dan})\) is derived from elements where \(d_{A}(\text{Angelo}) = 0.4\) and \(d_{B}(\text{Dan}) = 0.3\), resulting in \(d_{A \times B}(\text{Angelo}, \text{Dan}) = 0.4 \wedge 0.3 = 0.3\). This operation effectively maps how two fuzzy sets interact, providing insights into their combined characteristics.