Fuzzy Logic with Engineering Applications

Timothy Ross

Chapter 5 Logic and Fuzzy Systems - all with Video Answers

Educators

Chapter Questions

02:28

Problem 1

Under what conditions of $P$ and $Q$ is the implication $P \rightarrow Q$ a tautology?

J Hardin

Numerade Educator

04:30

The exclusive-or is given by the expression $P X O R Q=(\mathbb{P} \wedge Q) \vee(P \wedge \bar{Q})$. Show that the logical-or, given by $P \vee Q$, gives a different result from the exclusive-or and comment on this difference using an example in your own field.

Anas Venkitta

Numerade Educator

Problem 3

For a proposition $R$ of the form $\mathrm{P} \rightarrow \mathrm{Q}$, show the following:
(a) R and its contrapositive are equivalent, i.e., prove that $(\mathrm{P} \rightarrow \mathrm{Q}) \leftrightarrow(\overline{\mathrm{Q}} \rightarrow \overline{\mathrm{P}})$.
(b) The converse of $R$ and the inverse of $R$ are equivalent, i.e., prove that $(\mathrm{Q} \rightarrow \mathrm{P}) \leftrightarrow(\overline{\mathrm{P}} \rightarrow \overline{\mathrm{Q}})$.

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01:04

Problem 4

Show that the dual of the equivalence $((\mathrm{P} \vee \mathrm{Q}) \vee((\overline{\mathrm{P}}) \wedge(\overline{\mathrm{Q}}))) \leftrightarrow X$ is also true.

Christopher Stanley

Numerade Educator

Problem 5

Show that De Morgan's principles are duals.

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Problem 6

Show that the compound proposition $((\mathrm{P} \rightarrow \mathrm{Q}) \wedge(\mathrm{R} \rightarrow \overline{\mathrm{S}}) \wedge(\mathrm{Q} \rightarrow \mathrm{R})) \rightarrow(\mathrm{P} \rightarrow \overline{\mathrm{S}})$ is a tautology.

Dmitri Morenike

Numerade Educator

01:32

Problem 7

Show that the following propositions from Lewis Carroll are tautologies [Gill, 1976]:
(a) No ducks waltz; no officers ever decline to waltz; all my poultry are ducks. Therefore, none of my poultry are officers.
(b) Babies are illogical; despised persons cannot manage crocodiles; illogical persons are despised; therefore, babies cannot manage crocodiles.
(c) Promise-breakers are untrustworthy; wine-drinkers are very communicative; a man who keeps his promise is honest; all pawnbrokers are wine-drinkers; we can always trust a very communicative person; therefore, all pawnbrokers are honest. (This problem requires $2^6=64$ lines of a truth table; perhaps it should be tackled with a computer.)

Jiji Peter

Numerade Educator

Problem 8

Prove the following statements by contradiction.
(a) $((P \rightarrow Q) \wedge P) \rightarrow Q$
(b) $((\mathrm{P} \rightarrow \overline{\mathrm{Q}}) \wedge(\mathrm{Q} \vee \overline{\mathrm{R}}) \wedge(\mathrm{R} \wedge \overline{\mathrm{S}})) \rightarrow \overline{\mathrm{P}}$

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01:13

Problem 9

Prove that $((P \rightarrow \bar{Q}) \wedge(R \rightarrow \bar{Q}) \wedge(P \vee R)) \rightarrow R$ is not a tautology (i.e., a fallacy) by developing a counterexample.

Dalia Rodriguez

Numerade Educator

17:07

Problem 10

Prove that the following statements are tautologies.
(a) $((\mathrm{P} \rightarrow \mathrm{Q}) \wedge \mathrm{P}) \rightarrow \mathrm{Q}$
(b) $\mathrm{P} \rightarrow(\mathrm{P} \vee \mathrm{Q})$
(c) $(\mathrm{P} \wedge \mathrm{Q}) \rightarrow \mathrm{P}$
(d) $((\mathrm{P} \rightarrow \mathrm{Q}) \wedge(\mathrm{Q} \rightarrow \mathrm{R})) \rightarrow(\mathrm{P} \rightarrow \mathrm{R})$
(e) $((\mathrm{P} \vee \mathrm{Q}) \wedge \overline{\mathrm{P}}) \rightarrow \mathrm{Q}$

Sigurður Orri

Numerade Educator

Problem 11

For this inference rule,

$$
[(\mathrm{A} \rightarrow \mathrm{B}) \wedge(\mathrm{B} \rightarrow \mathrm{C})] \rightarrow(\mathrm{A} \rightarrow \mathrm{C})
$$

Prove that the rule is a tautology.

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Problem 12

Consider the following two discrete fuzzy sets, which are defined on universe $X=\{-5,5\}$ :

$$
\begin{aligned}
& \mathrm{A}={ }^{\prime \prime z e r o} "+=\left\{\frac{0}{-2}+\frac{0.5}{-1}+\frac{1.0}{0}+\frac{0.5}{1}+\frac{0}{2}\right\} \\
& \mathrm{B}=\text { "positive medium" }=\left\{\frac{0}{0}+\frac{0.5}{1}+\frac{1.0}{2}+\frac{0.5}{3}+\frac{0}{4}\right\}
\end{aligned}
$$
(a) Construct the relation for the rule IF A, THEN B (i.e., IF $x$ is "zero" THEN $y$ is "positive medium"') using the Mamdani implication, Eq. (5.20), and the product implication, Eq. (5.22), or

$$
\mu_{\mathrm{R}}(x, y)=\min \left[\mu_{\mathrm{A}}(x), \mu_{\mathrm{B}}(y)\right]
$$

and

$$
\mu_{\mathrm{R}}(x, y)=\mu_{\mathrm{A}}(x)-\mu_{\mathrm{B}}(y)
$$

(b) If we introduce a new antecedent,

$$
\underset{\sim}{\mathrm{A}^{\prime}}=\text { "'positive small" }=\left\{\frac{0}{-1}+\frac{0.5}{0}+\frac{1.0}{1}+\frac{0.5}{2}+\frac{0}{3}\right\}
$$

find the new consequent $\mathrm{B}^{\prime}$, using max $-\min$ composition, i.e., $\mathrm{B}^{\prime}=\mathrm{A}^{\prime} \circ \mathrm{R}$, for both relations from part (a).

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Problem 13

Given the fuzzy sets A and B on X and Y , respectively,

$$
\begin{aligned}
& A=\int\left\{\frac{1-0.1 x}{x}\right\}, \quad \text { for } x \in[0,+10] \\
& \underset{\sim}{B}=\int\left\{\frac{0.2 y}{y}\right\}, \quad \text { for } y \in[0,+5] \\
& \mu_{\mathrm{A}}(x)=0 \quad \text { outside the }[0,10] \text { interval } \\
& \mu_{\mathrm{B}}(y)=0 \quad \text { outside the }[0,5] \text { interval }
\end{aligned}
$$

(a) Construct a fuzzy relation R for the implication $\underset{\sim}{\mathrm{A}} \rightarrow \underset{\mathrm{B}}{ }$ using the classical implication operation, i.c., construct $R=(\mathrm{A} \times \mathrm{B}) \cup(\overline{\mathrm{A}} \times \mathrm{Y})$.
(b) Use max-min composition to find ${\underset{\sim}{B}}^{\prime}$, given

$$
\mathrm{A}^{\prime}=\left\{\frac{1}{3}\right\}
$$

Note: $\mathrm{A}^{\prime}$ is a crisp singleton, i.e., the number 3 has a membership of 1 , and all other numbers in the universe X have a membership of 0 .
Hint: You can solve this problem graphically by segregating the Cartesian space into various regions according to the min and max operations, or you can approximate the continuous fuzzy variables as discrete variables and use matrix operations. In either case, "sketch" the solutions for part (a) in 3D space $(x, y, \mu)$ and $(b)$ in 2D space $(y, \mu)$.

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Problem 14

Suppose we have a distillation process where the objective is to separate components of a mixture in the input stream. The process is pictured in Fig. P5.14. The relationship between the input variable, temperature, and the output variable, distillate fractions, is not precise but the human operator of this process has developed an intuitive understanding of this relationship. The universe for each of these variables is

$$
\begin{aligned}
& \mathrm{X}=\text { universe of temperatures }\left({ }^{\circ} \mathrm{F}\right)=\{160,165,170,175,180,185,190,195\} \\
& \mathrm{Y}=\text { universe of distillate fractions (percentages) }=\{77,80,83,86,89,92,95,98\}
\end{aligned}
$$
Now we define fuzzy sets $A$ and $B$ on $X$ and $Y$, respectively:

$$
\begin{aligned}
& \mathrm{A}=\text { temperature of input steam is hot }=\left\{\frac{0}{175}+\frac{0.7}{180}+\frac{1}{185}+\frac{0.4}{190}\right\} \\
& \mathrm{B}=\text { separation of mixture is good }=\left\{\frac{0}{89}+\frac{0.5}{92}+\frac{0.8}{95}+\frac{1}{98}\right\}
\end{aligned}
$$

We wish to determine the proposition, IF "temperature is hot"' THEN "separation of mixture is good,'' or symbolically, $\mathrm{A} \rightarrow \mathrm{B}$. From this,
(a) Find $\mathrm{R}=(\mathrm{A} \times \mathrm{B}) \cup(\overline{\mathrm{A}} \times \mathrm{Y})$.
(b) Now define another fuzzy linguistic variable as

$$
A^{\prime}=\left\{\frac{1}{170}+\frac{0.8}{175}+\frac{0.5}{180}+\frac{0.2}{185}\right\}
$$

and for the "new"' rula IF A' THEN B', find B' using max-min composition, i.c., find $\mathrm{B}^{\prime}=\mathrm{A}^{\prime} \circ \mathbf{R}$

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Problem 15

The calculation of the vibration of an elastic structure depends on knowing the material properties of the structure as well as its support conditions. Suppose we have an elastic structure, such as a bar of known material, with properties like wave speed ( $C$ ), modulus of elasticity $(E)$, and cross-sectional area ( $A$ ). However, the support stiffness is not well-known; hence the fundamental natural frequency of the system is not precise either. A relationship does exist between them, though, as illustrated in Fig. P5.15.
Define two fuzzy sets,
$\underset{\sim}{\mathrm{K}}=$ "support stiffness," in pounds per square inch
$\mathrm{f}_1=$ "first natural frequency of the system," in hertz
with membership functions

$$
\begin{aligned}
& \mathrm{K}=\left\{\frac{0}{1 e+3}+\frac{0.2}{1 e+4}+\frac{0.5}{1 e+5}+\frac{0.8}{5 e+5}+\frac{1}{1 e+6}+\frac{0.8}{5 e+6}+\frac{0.2}{1 e+7}\right\} \\
& \mathrm{f}_1=\left\{\frac{0}{100}+\frac{0}{200}+\frac{0.2}{500}+\frac{0.5}{800}+\frac{1}{1000}+\frac{0.8}{2000}+\frac{0.2}{5000}\right\}
\end{aligned}
$$

(a) Using the proposition, IF $x$ is K , THEN $y$ is $\mathrm{f}_1$, find this relation using the following forms of the implication $\underset{\sim}{\mathrm{K}} \rightarrow \underset{\text { fi }}{ }$ :
(i) Classical $\quad \mu_{\mathrm{R}}=\max \left[\min \left(\mu_{\mathrm{K}}, \mu_{\mathrm{f}_1}\right),\left(1-\mu_{\mathrm{K}}\right)\right]$
(ii) Mamdani $\mu_{\mathbb{R}}=\min \left(\mu_{\mathrm{K}}, \mu_{\mathrm{f}_{\mathrm{i}}}\right)$
(iii) Product $\quad \mu_{\mathbb{R}}=\mu_{\mathrm{K}} \cdot \mu_{\mathrm{f}_{\mathrm{I}}}$
(b) Now define another antecedent, say $\mathrm{K}^{\prime}=$ "damaged support,"

$$
\mathrm{K}^{\prime}=\left\{\frac{0}{1 e+3}+\frac{0.8}{1 e+4}+\frac{0.1}{1 e+5}\right\}
$$

Find the system's fundamental (first) natural frequency due to the change in the support conditions, i.e., find $f_1=$ "first natural frequency due to damaged support" using classical implication from part (a), sub-part (i) preceding, and
(i) $\max -\mathrm{min}$ composition
(ii) max-product composition

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Problem 16

When gyros are calibrated for axis bias, they are matched with a temperature. Thus, we can have a relation of gyro bias (GB) vs. temperature (T). Suppose we have fuzzy sets for a given gyro bias and a given Fahrenheit temperature, as follows:

$$
\begin{aligned}
& \mu_{\mathrm{GB}}(x)=\left\{\frac{0.2}{3}+\frac{0.4}{4}+\frac{1}{5}+\frac{0.4}{6}+\frac{0.2}{7}\right\} \text { bias in degrees Fahrenheit per hour } \\
& \mu_{\underline{\mathrm{T}}}(y)=\left\{\frac{0.4}{66}+\frac{0.6}{68}+\frac{1}{70}+\frac{0.6}{72}+\frac{0.4}{74}\right\} \text { temperature in degrees Fahrenheit }
\end{aligned}
$$

(a) Use a Mamdani implication to find the relation IF gyro bias, THEN temperature.
(b) Suppose we are given a new gyro bias ( $\left.\mathrm{GB}^{\prime}\right)$ as follows:

$$
\mu_{\mathrm{GBB}^{\prime}}(x)=\left\{\frac{0.6}{3}+\frac{1}{4}+\frac{0.6}{5}\right\}
$$

Using max-min composition, find the temperature associated with this new bias.

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Problem 17

You are asked to develop a controller to regulate the temperature of a room. Knowledge of the system allows you to construct a simple rule of thumb: when the temperature is HOT then cool room down by turning the fan at the fast speed, or, expressed in rule form, IF temperature is HOT, THEN fan should turn FAST. Fuzzy sets for hot temperature and fast fan speed can be developed: for example,

$$
\underset{\sim}{\mathrm{H}}=\text { "hot" }^{\prime}=\left\{\frac{0}{60}+\frac{0.1}{70}+\frac{0.7}{80}+\frac{0.9}{90}+\frac{1}{100}\right\}
$$
represents universe X in ${ }^{\circ} \mathrm{F}$, and
represents universe Y in 1000 rpm .
(a) From these two fuzzy sets construct a relation for the rule using classical implication.
(b) Suppose a new rule uses a slightly different temperature, say "moderately hot," and is expressed by the fuzzy membership function for "moderately hot," or

$$
\mathrm{H}^{\prime}=\left\{\frac{0}{60}+\frac{0.2}{70}+\frac{1}{80}+\frac{1}{90}+\frac{1}{100}\right\}
$$

Using max-product composition, find the resulting fuzzy fan speed.

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05:53

Problem 18

In public transportation systems there often is a significant need for speed control. For subway systems, for example, the train speed cannot go too far beyond a certain target speed or the trains will have trouble stopping at a desired location in the station. Set up a fuzzy set

$$
A={ }^{\prime} \text { speed way over target" }=\left\{\frac{0}{T_0}+\frac{0.8}{T_0+5}+\frac{1}{T_0+10}+\frac{0.8}{T_0+15}\right\}
$$

on a universe of target speeds, say $\mathrm{T}=\left[T_0, T_0+15\right]$, where $T_0$ is a lower bound on speed. Define another fuzzy set,

$$
\underset{\sim}{\mathrm{B}}=\text { "'apply brakes with high force"' }=\left\{\frac{0.3}{10}+\frac{0.8}{20}+\frac{0.9}{30}+\frac{1}{40}\right\}
$$

on a universe of braking pressures, say $\mathrm{S}=[10,40]$.
(a) For the compound proposition, IF speed is "way over target," THEN "apply brakes with high force," find a fuzzy relation using classical implication.
(b) For a new antecedent,

$$
\mathrm{A}^{\prime}={ }^{\prime \prime} \text { speed moderately over target }{ }^{\prime}=\left\{\frac{0.2}{T_0}+\frac{0.6}{T_0+5}+\frac{0.8}{T_0+10}+\frac{0.3}{T_0+15}\right\}
$$

find the fuzzy brake pressure using max-min composition.

Lucas Finney

Numerade Educator

Problem 19

We want to consider the engineering of amplifiers. Here, the amplifier is a simple voltagemeasuring input and current output, as shown in Fig. P5.19. We define two fuzzy linguistic variables for a fuzzy relation: $\underline{V}_{ \pm}$, the input voltage, and $\underline{I}_{\text {out }}$, the output current:

$$
\begin{aligned}
& \mathrm{V}_{\mathrm{s}}={ }^{\prime \text { small" }}=\left\{\frac{0.5}{0.10}+\frac{1}{0.20}+\frac{0.8}{0.30}+\frac{0.2}{0.40}\right\} \quad \text { volts } \\
& \mathrm{I}_{\text {oat }}={ }^{\prime \text { big} "}=\left\{\frac{0.3}{0.6}+\frac{1}{1}+\frac{0.5}{1.4}\right\} \quad \mathrm{amps}
\end{aligned}
$$

where $\mathrm{V}_{\text {in }}$ is defined on a universe of voltages, and $\mathrm{I}_{\text {out }}$ is defined on a universe of currents.
(a) Find the relation, IF $\mathrm{V}_{\mathrm{in}}$, THEN $\underline{I}_{\text {gut }}$, using classical implication.
(b) Another fuzzy linguistic variable in this problem is input impedance, $Z$. The higher the impedance, generally the better the amplifier. For the following impedance defined on a universe of resistances,

$$
\underset{\sim}{Z}=\text { "high impedance" }=\left\{\frac{0}{10^4}+\frac{0.3}{10^5}+\frac{1}{10^6}+\frac{0.6}{10^7}\right\} \quad \text { ohms }
$$

find the relation, IF Z, THEN $\underline{I}_{\text {out }}$, using Mamdani implication.

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Problem 20

For Example 5.9 in this chapter, recalculate the fuzzy relation R using
(a) Equation (5.19)
(b) Equation (5.20)
(c) Equation (5.21)
(d) Equation (5.22)
(e) Equation (5.23)

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Problem 21

Fill in the following table using Eqs. (5.19) - (5.23) to determine the values of the implication $\mathrm{A} \rightarrow$ B. Comment on the similarities and dissimilarities of the various implication methods with respect to the various values for $\underset{\sim}{\mathrm{A}}$ and B .

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Problem 22

A factory process control operation involves two linguistic (atomic) parameters consisting of pressure and temperature in a fluid delivery system. Nominal pressure limits range from 400 psi minimum to 1000 psi maximum. Nominal temperature limits are 130 to $140^{\circ} \mathrm{F}$. We characterize each parameter in fuzzy linguistic terms as follows:

$$
\begin{aligned}
& \text { "Low temperature" }=\left\{\frac{1}{131}+\frac{0.8}{132}+\frac{0.6}{133}+\frac{0.4}{134}+\frac{0.2}{135}+\frac{0}{136}\right\} \\
& { }^{\text {"High temperature" }}=\left\{\frac{0}{134}+\frac{0.2}{135}+\frac{0.4}{136}+\frac{0.6}{137}+\frac{0.8}{138}+\frac{1}{139}\right\} \\
& { }^{\text {"High pressure" }}=\left\{\frac{0}{400}+\frac{0.2}{600}+\frac{0.4}{700}+\frac{0.6}{800}+\frac{0.8}{900}+\frac{1}{1000}\right\} \\
& \text { "Low pressure" }=\left\{\frac{1}{400}+\frac{0.8}{600}+\frac{0.6}{700}+\frac{0.4}{800}+\frac{0.2}{900}+\frac{0}{1000}\right\}
\end{aligned}
$$

(a) Find the following membership functions:
(i) Temperature not very low
(ii) Temperature not very high
(iii) Temperature not very low and not very high
(b) Find the following membership functions:
(i) Pressure slightly high
(ii) Pressure fairly high ([high $]^{2 / 3}$ )
(iii) Pressure not very low or fairly low

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Problem 23

In information retrieval, having fuzzy information about the size of a document helps when trying to scale the word frequencies of the document (i.e., how often a word occurs in a document is important when determining relevance). So on the universe of document sizes, we define two fuzzy sets:

$$
\begin{aligned}
& \text { "'Small'' document" }= \begin{cases}1-e^{-k(a-x)} & \text { for } x \leq a \\
0 & \text { for } x>a\end{cases} \\
& \text { "Large document" }= \begin{cases}1-e^{-k(x-b)} & \text { for } x \geq b \\
0 & \text { for } x<b\end{cases}
\end{aligned}
$$

where the parameters $k, a$, and $b$ change from database to database. Graphically the parameters $a$ and $b$ look as shown in Fig. P5.23. Develop a graphical solution to the following linguistic phrases, for the specific values of $a=2, b=4$, and $k=0.5$ :
(a) "Not very large" document
(b) 'Large and small'' documents
(c) "Not very large or small" documents

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Problem 24

In a problem related to the computer tracking of soil particles as they move under stress, the program displays desired particles on the screen. Particles can be small and large. Because of segmentation problems in computer imaging, the particles can become too large and obscure particles of interest or become too small and be obscured. To solve this problem linguistically, suppose we define the following atomic terms on a scale of sizes $[0,50]$ in units of $\mathrm{mm}^2$ :

$$
\begin{aligned}
& " \text { Large" }=\left\{\frac{0}{0}+\frac{0.1}{10}+\frac{0.3}{20}+\frac{0.5}{30}+\frac{0.6}{40}+\frac{0.7}{50}\right\} \\
& \text { "Small" }^{\prime}=\left\{\frac{1}{0}+\frac{0.8}{10}+\frac{0.5}{20}+\frac{0.3}{30}+\frac{0.1}{40}+\frac{0}{50}\right\}
\end{aligned}
$$

For these atomic terms find membership functions for the following phrases:
(a) Very small or very large
(b) Not small and not large
(c) Large or not small

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02:14

Problem 25

In vehicle navigation the mapping source of information uses shape points to define the curvature of a turning maneuver. A segment is a length of road between two points. If the segment is linear, it has no or very few shape points. If the road is winding or circular, the segment can have many shape points. Figure P5. 25 shows the relationship of curvature and shape points. Assume that up to nine shape points can define any curvature in a typical road segment. The universe of discourse of shape points then varies from 0 (linear road) to 9 (extremely curved). Define the following membership functions:

$$
\begin{aligned}
\text { "Somewhat straight" } & =\left\{\frac{1}{0}+\frac{0.9}{1}+\frac{0.8}{2}+\frac{0.7}{3}+\frac{0.6}{4}+\frac{0.5}{5}+\frac{0.4}{6}+\frac{0.3}{7}+\frac{0.2}{8}+\frac{0.1}{9}\right\} \\
\text { "Curved" } & =\left\{\frac{0}{0}+\frac{0.1}{1}+\frac{0.2}{2}+\frac{0.3}{3}+\frac{0.4}{4}+\frac{0.5}{5}+\frac{0.6}{6}+\frac{0.7}{7}+\frac{0.8}{8}-\frac{0.9}{9}\right\}
\end{aligned}
$$
Calculate the membership functions for the following phrases:
(a) Very curved
(b) Fairly curved ( $=[\text { curved }]^{2 / 3}$ )
(c) Very, very somewhat straight
(d) Not fairly curved and very, very somewhat straight

Eric Mockensturm

Numerade Educator

Problem 26

This problems deals with the voltages generated internally in switching power supplies. Embedded systems are often supplied 120 V AC for power. A "power supply"' is required to convert this to a useful voltage (quite often +5 V DC ). Some power supply designs employ a technique called "switching." This technique generates the appropriate voltages by storing and releasing the energy between inductors and capacitors. This problem characterizes two linguistic variables, high and low voltage, on the voltage range of 0 to 200 VAC :

$$
\begin{aligned}
\text { 'High" }^{\prime} & =\left\{\frac{0}{0}+\frac{0}{25}+\frac{0}{50}+\frac{0.1}{75}+\frac{0.2}{100}+\frac{0.4}{125}+\frac{0.6}{150}+\frac{0.8}{175}+\frac{1}{200}\right\} \\
\text { "Medium" }^{\prime} & =\left\{\frac{0.2}{0}+\frac{0.4}{25}+\frac{0.6}{50}+\frac{0.8}{75}+\frac{1}{100}+\frac{0.8}{125}+\frac{0.6}{150}+\frac{0.4}{175}+\frac{0.2}{200}\right\}
\end{aligned}
$$

Find the membership functions for the following phrases:
(a) Not very high
(b) Slightly medium and very high
(c) Very, very high or very, very medium

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Problem 27

In risk assessment we deal with characterizing uncertainty in assessing the hazard to human health posed by various toxic chemicals. Because the pharmacokinetics of the human body are very difficult to explain for long-term chemical hazards, such as chronic exposure to lead or to cigarette smoke, hazards can sometimes be uncertain because of scarce data or uncertainty in the exposure patterns. Let us characterize hazard linguistically with two terms: "low" hazard and "high" hazard:

$$
\begin{aligned}
& \text { "Low" hazard }=\left\{\frac{0}{1}+\frac{0.3}{2}+\frac{0.8}{3}+\frac{0.1}{4}+\frac{0}{5}\right\} \\
& \text { "High" hazard }=\left\{\frac{0}{1}+\frac{0.1}{2}+\frac{0.2}{3}+\frac{0.8}{4}+\frac{0}{5}\right\}
\end{aligned}
$$

Find the membership functions for the following linguistic expressions:
(a) Low hazard and not high hazard
(b) Very high hazard and not low hazard
(c) Low hazard or high hazard

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08:29

Problem 28

In reference to car speeds we have the linguistic variables "fast" and "slow" for speed:

$$
\begin{aligned}
& \text { "Fast'" }^{\prime}=\left\{\frac{0}{0}+\frac{0.1}{10}+\frac{0.2}{20}+\frac{0.3}{30}+\frac{0.4}{40}+\frac{0.5}{50}+\frac{0.6}{60}+\frac{0.7}{70}+\frac{0.8}{80}+\frac{0.9}{90}+\frac{1}{100}\right\} \\
& \text { "Slow'" }^{\prime}=\left\{\frac{1}{0}+\frac{0.9}{10}+\frac{0.8}{20}+\frac{0.7}{30}+\frac{0.6}{40}+\frac{0.5}{50}+\frac{0.4}{60}+\frac{0.3}{70}+\frac{0.2}{80}+\frac{0.1}{90}+\frac{0}{100}\right\}
\end{aligned}
$$

Using these variables, compute the membership function for the following linguistic terms:
(a) Very fast
(b) Very, very fast
(c) Highly fast (= minus very, very fast)
(d) Plus very fast
(e) Fairly fast $\left(=[\text { fast }]^{2 / 3}\right)$
( $f$ ) Not very slow and not very fast
(g) Slow or not very slow

Kaitlin Coad

Numerade Educator

00:26

Problem 29

For finding the volume of a cylinder, we need two parameters, namely, radius and height of the cylinder. When the radius is 7 centimeters and height is 12 centimeters, then the volume equals 1847.26 cubic centimeters (using volume $=\pi r^2 h$ ). Reduce the following rule to canonical form: IF $x_1$ is radius AND $x_2$ is height, THEN $y$ is volume.

Amy Jiang

Numerade Educator

01:19

Problem 30

According to Boyle's law, for an ideal gas at constant temperature $t$, pressure is inversely proportional to volume, or volume is inversely proportional to pressure. When we consider different sets of pressures and volumes under the same temperature, we can apply the following rule: IF $x_1$ is $p_1 v_1$ AND $x_2$ is $p_2 v_2$, THEN $t$ is a constant. Here $p$ is pressure and $v$ is volume of the gas considered. Reduce this rule to canonical form.

Linh Vu

Numerade Educator

Problem 31

In Example 5.16 recalculate the response function shown in Fig. $5.15 b$ using the following membership function shapes:
(a) two triangles for the input and two triangles for the output;
(b) two trapezoids for the input and two trapezoids for the output

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Problem 32

In Example 5.17 recalculate the response function shown in Fig. $5.17 d$ using the following membership function shapes for the inputs: (a) triangles for small, medium, large; (b) trapezoids for small, medium, large.

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Problem 33

Repeat Example 5.16 using a weighted sum defiuzzifier instead of the weighted average defuzzification, i.e., use $z=w_1 z_1+w_2 z_2$ in Fig. 5.14. Do you get the same response surface as in Example 5.16? Why?

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05:50

Problem 34

From thermodynamics it is known that for an ideal gas in an adiabatic reversible process

$$
\frac{T_2}{T_1}=\left(\frac{P_2}{P_1}\right)^{\frac{\gamma-1}{\gamma}}
$$

where $T_1$ and $T_2$ are temperatures in kelvin $(\mathrm{k})$ and $P_1$ and $P_2$ are pressures in bars and, for an ideal gas, the constant $\gamma$ is

$$
\gamma=1+\frac{R}{C v}=1.4
$$

For this problem, $T_1$ will be fixed at 300 K and the fuzzy model will predict $P_2$ for the given input variables $P_1$ and $T_2$. In other words, we are interested in finding the final pressure, $P_2$, of the system if the temperature of the system is changed to $T_2$ from an original pressure equal to $P_1$. A real application could use a similar model built from experimental data to do a prediction on nonideal gases.

The rules used are
Rule 1: IF $P_1=\operatorname{atmP}$ AND $T_2=$ lowT THEN $P_2=$ lowP
Rule 2: IF $P_1=\operatorname{atmP}$ AND $T_2=$ midT THEN $P_2=$ lowP
Rule 1: IF $P_1=$ lowP AND $T_2=$ lowT THEN $P_2=$ very highP
The rules and membership functions are based on values obtained from the known formula where

IF $P_1=1$ bar AND $T_2=410 \mathrm{~K}$ THEN $P_2=3 \mathrm{bar}$
IF $P_1=1$ bar AND $T_2=430 \mathrm{~K}$ THEN $P_2=3.5$ bar
IF $P_1=2$ bar AND $T_2=420 \mathrm{~K}$ THEN $P_2=6 \mathrm{bar}$
Given the rule-base, the membership functions shown in Fig. P5.34, and the following pair of input values, $P_1=1.6$ bar and $T_2=415 \mathrm{~K}$, conduct a simulation to determine $P_2$ for the three inference methods of Mamdani, Sugeno, and Tsukamoto. For the Sugeno consequents use the ideal gas formula, given above.

For Mamdani and Sugeno, use the input membership functions in Fig. P5.34a.
For the Tsukamoto method, use the output membership functions shown in Fig. P5.34b with the same inputs as used in Fig. P5.34a.

Lottie Adams

Numerade Educator

Problem 35

In finding the Nusselt number (a dimensionless number for determining heat transfer) for an hexagonal cylinder in cross flow, there are two correlations (which are to be used as the consequent terms in a Sugeno inference method):

$$
\begin{aligned}
& \mathrm{Nu}_1=0.16 \mathrm{Re}^{0.633} \operatorname{Pr}^{1 / 3} \quad 5000<\operatorname{Re}<19,650 \\
& \mathrm{Nu}_2=0.0385 \operatorname{Re}^{0.728} \operatorname{Pr}^{1 / 3} \operatorname{Re}>19,650
\end{aligned}
$$

Re is the Reynolds number and $\operatorname{Pr}$ is the Prandtl number.
The Nusselt number is a function of convective heat transfer $(h)$, diameter of the hexagonal cylinder $(D)$ over which cooling fluid travels, and the conductivity of the material $(K)$ :

$$
\mathrm{Nu}=\frac{h D}{K}
$$
Both Re and Pr can be fuzzy due to uncertainty in the variables in velocity. It would be convenient to find Nu (output) based on Re and Pr (inputs) without having to do all the calculations. More specifically, there is uncertainty in calculating the Reynolds number because velocity is not known exactly:

$$
\operatorname{Re}=\frac{\rho V D}{\mu}
$$

where $\rho$ is the density, $V$ is the velocity, $D$ is the characteristic length (or pipe diameter), and $\mu$ is the dynamic viscosity. And there is also uncertainty in the value for the Prandtl number due to its constituents

$$
\operatorname{Pr}=\frac{v}{\alpha}
$$

where $v$ is the kinematic viscosity and $\alpha$ is the specific gravity.
Calculation of Nu is very involved and the incorporation of a rule-base can be used to bypass these calculations; we have the following rules to govern this process:

If Re is high and Pr is low $\longrightarrow$ Then Nu is low
If Re is low and Pr is low $\longrightarrow$ Then Nu is low
If Re is high and Pr is high $\longrightarrow$ Then Nu is medium
If Re is low and Pr is high $\longrightarrow$ Then Nu is medium
For this problem, conduct a Mamdani and a Sugeno inference, based on the membership functions given in Figs P5.35a, $b$, and $c$, and use the following inputs:

$$
\begin{aligned}
\mathrm{Re} & =19.65 \times 10^3 \\
\operatorname{Pr} & =275
\end{aligned}
$$

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