Question

Let the membership functions of fuzzy sets \(F\) and \(G\) be asfollows: $$ X: 11,2,3,4,5,6,7,8,9,10\\} $$ $$ \begin{aligned} &F:[0,0,0,0,0,1,0.8,0.5,0.9,1,1] \\ &G:[0,0,0,1,0.5,0.9,1,0.9,0.5,0.0] \end{aligned} $$ State whether \((\) i) \(F=G\) (ii) \(F\) is a subset of \(G\). Also write down \(F^{\circ}, P \cap G\) and \(F \cup G\).

Short answer

F ≠ G; F is not a subset of G. Complement: [1, 1, 1, 1, 1, 0, 0.2, 0.5, 0.1, 0]. Intersection: [0, 0, 0, 0, 0, 0.9, 0.8, 0.5, 0.5, 0]. Union: [0, 0, 0, 1, 0.5, 1, 1, 0.9, 0.9, 1].

Step-by-step solution

Determining Equality (F = G)

Two fuzzy sets are equal if they have the same membership degree across all elements. Let's compare the membership functions of sets \(F\) and \(G\) point by point for each element in the universe of discourse \(X\):- \(F(1) = 0\) vs. \(G(1) = 0\)- \(F(2) = 0\) vs. \(G(2) = 0\)- \(F(3) = 0\) vs. \(G(3) = 0\)- \(F(4) = 0\) vs. \(G(4) = 1\)- \(F(5) = 0\) vs. \(G(5) = 0.5\)- \(F(6) = 1\) vs. \(G(6) = 0.9\)- \(F(7) = 0.8\) vs. \(G(7) = 1\)- \(F(8) = 0.5\) vs. \(G(8) = 0.9\)- \(F(9) = 0.9\) vs. \(G(9) = 0.5\)- \(F(10) = 1\) vs. \(G(10) = 0\)Since \(F(x) eq G(x)\) for any \(x\) where their membership values differ, \(F eq G\).

Checking Subset (F ⊆ G)

A fuzzy set \(F\) is a subset of \(G\) if for all elements in \(X\), the membership degree of \(F\) is less than or equal to \(G\):- \(F(1) \leq G(1)\): 0 ≤ 0- \(F(2) \leq G(2)\): 0 ≤ 0- \(F(3) \leq G(3)\): 0 ≤ 0- \(F(4) \leq G(4)\): 0 ≤ 1- \(F(5) \leq G(5)\): 0 ≤ 0.5- \(F(6) \leq G(6)\): 1 ≤ 0.9 (False)- \(F(7) \leq G(7)\): 0.8 ≤ 1- \(F(8) \leq G(8)\): 0.5 ≤ 0.9- \(F(9) \leq G(9)\): 0.9 ≤ 0.5 (False)- \(F(10) \leq G(10)\): 1 ≤ 0 (False)Since there are values where \(F(x) > G(x)\), \(F\) is not a subset of \(G\).

Complement of F (F°)

The complement of a fuzzy set \(F\), denoted \(F^\circ\), is given by \(1 - \text{the membership value for each element in } F\).Thus, \(F^\circ\) is:- \(1 - F(1) = 1 \)- \(1 - F(2) = 1 \)- \(1 - F(3) = 1 \)- \(1 - F(4) = 1 \)- \(1 - F(5) = 1 \)- \(1 - F(6) = 0 \)- \(1 - F(7) = 0.2\)- \(1 - F(8) = 0.5\)- \(1 - F(9) = 0.1\)- \(1 - F(10) = 0 \)Thus, \(F^\circ = [1, 1, 1, 1, 1, 0, 0.2, 0.5, 0.1, 0]\).

Intersection of F and G (F ∩ G)

The intersection is determined by taking the minimum of the membership values of \(F\) and \(G\) for each element:\(F ∩ G:\)- \(\min(0, 0) = 0\)- \(\min(0, 0) = 0\)- \(\min(0, 0) = 0\)- \(\min(0, 1) = 0\)- \(\min(0, 0.5) = 0\)- \(\min(1, 0.9) = 0.9\)- \(\min(0.8, 1) = 0.8\)- \(\min(0.5, 0.9) = 0.5\)- \(\min(0.9, 0.5) = 0.5\)- \(\min(1, 0) = 0\)Thus, \(F ∩ G = [0, 0, 0, 0, 0, 0.9, 0.8, 0.5, 0.5, 0]\).

Union of F and G (F ∪ G)

The union is determined by taking the maximum of the membership values of \(F\) and \(G\) for each element:\(F ∪ G:\)- \(\max(0, 0) = 0\)- \(\max(0, 0) = 0\)- \(\max(0, 0) = 0\)- \(\max(0, 1) = 1\)- \(\max(0, 0.5) = 0.5\)- \(\max(1, 0.9) = 1\)- \(\max(0.8, 1) = 1\)- \(\max(0.5, 0.9) = 0.9\)- \(\max(0.9, 0.5) = 0.9\)- \(\max(1, 0) = 1\)Thus, \(F ∪ G = [0, 0, 0, 1, 0.5, 1, 1, 0.9, 0.9, 1]\).

Key concepts

Membership Functions

In the realm of fuzzy sets, a membership function plays a pivotal role. It acts as a bridge, defining how each element in a set belongs to the fuzzy set. Unlike classical sets where an element is either in or out (with values of 0 or 1), fuzzy set membership is measured on a scale from 0 to 1.

For instance, the membership function of a fuzzy set \(F\) could be expressed as \(F(x) = ext{membership value}\), where \('x'\) is an element from the universe of discourse. If \(x = 6\) in set \(F\) has a membership value of 1, it means this element fully belongs to \(F\). Conversely, a membership value closer to 0 indicates lesser degree of belonging.

• Fuzzy Array: Numbers represent how much each element belongs to the set.
• Range: Value runs from 0 (not a member) to 1 (complete member).
• Example: \(F = [0, 0, 0, 0, 0, 1, 0.8, 0.5, 0.9, 1]\)

Complement of a Fuzzy Set

The complement of a fuzzy set is an interesting concept where essentially the degree of membership is inverted. This is denoted as \(F^{\circ}\), calculated through subtracting each element's membership degree from 1.

To compute the complement, consider a fuzzy set \(F\) with membership degrees. For an element \(x\) within \(F\), its complement will be \(1 - F(x)\). The idea is simple but powerful: higher membership in the original set means lesser membership in the complement, and vice versa.

• Formula: \(F^{\circ}(x) = 1 - F(x)\)
• Inverse Relationship: If \(F(x) = 1\), then \(F^{\circ}(x) = 0\)
• Example: If \(F = [0, 0, 0, 0, 0, 1, 0.8, 0.5, 0.9, 1]\), then \(F^{\circ} = [1, 1, 1, 1, 1, 0, 0.2, 0.5, 0.1, 0]\)

Intersection and Union of Fuzzy Sets

Intersection and union are two core operations in fuzzy set theory, allowing us to combine sets in meaningful ways. In classical set theory, intersection identifies common elements, while union combines all elements. Fuzzy sets generalize this into membership functions by selecting the minimum (for intersections) or maximum (for unions) membership degrees.

Intersection of Fuzzy Sets

To find the intersection, represented as \(F \cap G\), use the minimum function on membership degrees. This indicates the smallest degree to which an element is a member of both sets. If an element in set \(F\) and set \(G\) has membership values of 0.5 and 0.7 respectively, the intersection degree is \(\min(0.5, 0.7) = 0.5\).

• Definition: \(F \cap G(x) = \min(F(x), G(x))\)
• Example: From initial sets, \(F \cap G = [0, 0, 0, 0, 0, 0.9, 0.8, 0.5, 0.5, 0]\)

Union of Fuzzy Sets

Conversely, union of fuzzy sets, denoted \(F \cup G\), involves picking the greater membership value from each set for corresponding elements. This shows where either set possesses significant membership. For instance, elements with memberships 0.2 in \(F\) and 0.4 in \(G\) will have a union membership of \(\max(0.2, 0.4) = 0.4\).

• Definition: \(F \cup G(x) = \max(F(x), G(x))\)
• Example: For given sets, \(F \cup G = [0, 0, 0, 1, 0.5, 1, 1, 0.9, 0.9, 1]\)