Appendix D: Defining Boolean and Fuzzy Logic Operators

• Val Lowndes
Chapter
Part of the Simulation Foundations, Methods and Applications book series (SFMA)

Abstract

Definition boolean logic

19.1 Definition Boolean Logic

$$\begin{array}{*{20}l} {{\text{If an element }}x} \hfill & {} \hfill \\ {\quad \quad \quad {\text{is contained in set }}A{\text{ then}}} \hfill & {\mu_{A} (x) = 1\quad \text{if}\quad x \in A} \hfill \\ {\quad \quad \quad {\text{is not contained in set }}A{\text{ then}}} \hfill & {\mu_{A} (x) = 0} \hfill \\ {\text{As}} \hfill & {\mu_{A\mathop \cup \nolimits B} (x) = 1\quad \text{if}\quad x \in A\;\text{or}\;x \in B} \hfill \\ {\text{and}} \hfill & {\mu_{A\mathop \cap \nolimits B} (x) = 1\quad \text{if}\quad x \in A\; \text{and}\;x \in B} \hfill \\ {\text{Then it follows that}} \hfill & {A \cup B \to \mu_{A \cup B} (x) = \hbox{max} \left[ {\mu_{A} \left( x \right),\mu_{B} (x)} \right]} \hfill \\ {\text{and}} \hfill & {A \cap B \to \mu_{A \cap B} (x) = \hbox{min} \left[ {\mu_{A} \left( x \right),\mu_{B} (x)} \right]} \hfill \\ \end{array}$$

19.2 Definition Fuzzy Logic

\begin{aligned} & \begin{array}{*{20}l} {{\text{If an element }}x} \hfill & {} \hfill \\ {\quad \quad \quad {\text{has some membership in set }}A{\text{ then}}} \hfill & {\mu_{A} \left( x \right) = k_{A} \quad 0 < k_{A} \le 1} \hfill \\ {\quad \quad \quad {\text{has no membership in set }}A{\text{ then}}} \hfill & {\mu_{A} (x) = 0} \hfill \\ {\text{define}} \hfill & {A \cup B \to \mu_{A \cup B} (x) = \hbox{max} \left[ {\mu_{A} \left( x \right),\mu_{B} (x)} \right]} \hfill \\ {\text{and}} \hfill & {A \cap B \to \mu_{A \cap B} (x) = \hbox{min} \left[ {\mu_{A} \left( x \right),\mu_{B} (x)} \right]} \hfill \\ \end{array} \\ & \begin{array}{*{20}c} {\text{thus}} & {\begin{array}{*{20}l} {\quad \quad \quad 0 \le \mu_{A \cup B} (x) \le 1} \hfill & {\text{and}} \hfill & {0 \le \mu_{A \cap B} (x) \le 1} \hfill \\ \end{array} } \\ \end{array} \\ \end{aligned}