Question

The complement of a fuzzy set S is the set\(\mathop S\limits^\_ \), with the degree of the membership of an element \(\mathop S\limits^\_ \)equal to 1 minus the degree of membership of this element in S. Find \(\mathop F\limits^\_ \) (the fuzzy set of people who are not famous) and \(\mathop R\limits^\_ \) (the fuzzy set of people who are not rich).

Short answer

It is obtained that, \(\mathop F\limits^\_ = \left\{ {\left( {0.6,Alice} \right),\left( {0.9,Brian} \right),\left( {0.4,Fred} \right),\left( {0.1,Oscar} \right),\left( {0.9,Rita} \right)} \right\}\)and

\(\mathop R\limits^\_ = \left\{ {\left( {0.4,Alice} \right),\left( {0.8,Brian} \right),\left( {0.2,Fred} \right),\left( {0.9,Oscar} \right),\left( {0.7,Rita} \right)} \right\}\).

Step-by-step solution

Explanation

In the given question,

Given in the question,

F = Set of all famous people.

\(F = \left\{ {\left( {0.6,Alice} \right),\left( {0.9,Brian} \right),\left( {0.4,Fred} \right),\left( {0.1,Oscar} \right),\left( {0.9,Rita} \right)} \right\}\)

R = Set of all reach people.

\(R = \left\{ {\left( {0.4,Alice} \right),\left( {0.8,Brian} \right),\left( {0.2,Fred} \right),\left( {0.9,Oscar} \right),\left( {0.7,Rita} \right)} \right\}\)

Concept

\(\mathop R\limits^\_ \)={ Degree of membership of an element in \(\mathop R\limits^\_ \) is equal to 1minus degree of membership of this element in S}

\(\mathop F\limits^\_ = \){ Degree of membership of an element in \(\mathop F\limits^\_ \)is equal to 1minus degree of membership of this element in S}

solution

Here we solve,

\(\mathop F\limits^\_ \)= Set of not famous people.

\(\mathop F\limits^\_ = \left\{ {\left( {1 - 0.4,Alice} \right),\left( {1 - 0.8,Brian} \right),\left( {1 - 0.2,Fred} \right),\left( {1 - 0.9,Oscar} \right),\left( {1 - 0.7,Rita} \right)} \right\}\)

\( = \left\{ {\left( {0.6,Alice} \right),\left( {0.9,Brian} \right),\left( {0.4,Fred} \right),\left( {0.1,Oscar} \right),\left( {0.9,Rita} \right)} \right\}\)

\(\mathop R\limits^\_ \)= Set of not-rich people.

\(\begin{aligned}{l}\mathop R\limits^\_ = \left\{ {\left( {1 - 0.6,Alice} \right),\left( {1 - 0.9,Brian} \right),\left( {1 - 0.4,Fred} \right),\left( {1 - 0.1,Oscar} \right),\left( {1 - 0.9,Rita} \right)} \right\}\\ = \left\{ {\left( {0.4,Alice} \right),\left( {0.8,Brian} \right),\left( {0.2,Fred} \right),\left( {0.9,Oscar} \right),\left( {0.7,Rita} \right)} \right\}\end{aligned}\)