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Fuzzy Logic with Engineering Applications

Timothy Ross

Chapter 2

Classical Sets and Fuzzy Sets - all with Video Answers

Educators

Chapter Questions

01:32

Problem 1

Typical membership functions for laminar and turbulent flow for a flat plate with a sharp leading edge in a typical air stream are shown in Fig. P2.1. Transition between laminar and turbulent flow usually takes place between Reynolds numbers of $2 \times 10^5$ and $3 \times 10^6$. An $\operatorname{Re}=5 \times 10^5$ is usually considered the point of turbulent flow for this situation. Find the intersection and union for the two flows.

Dominador Tan
Dominador Tan
Numerade Educator

Problem 2

In neighborhoods there may be several storm-water ponds draining to a single downstream trunk sewer. In this neighborhood the city monitors all ponds for height of water caused by storm events. For two storms (labeled A and B) identified as being significant based on rainfall data collected at the airport, determine the corresponding performance of the neighborhood storm-water ponds.
Suppose the neighborhood has five ponds, i.e., $\mathrm{X}=[1,2,3,4,5]$, and suppose that significant pond storage membership is 1.0 for any pond that is $70 \%$ or more to full depth. For storm A the pond performance set is

$$
\mathrm{A}=\left\{\frac{0.6}{1}+\frac{0.3}{2}+\frac{0.9}{3}+\frac{1}{4}+\frac{1}{5}\right\}
$$

For storm $\underset{\sim}{\mathrm{B}}$ the pond performance set is

$$
\underset{\sim}{\mathrm{B}}=\left\{\frac{0.8}{1}+\frac{0.4}{2}+\frac{0.9}{3}+\frac{0.7}{4}+\frac{1}{5}\right\}
$$

(a) To assess the impacts on pond performance suppose only two ponds can be monitored due to budget constraints. Moreover, data from the storms indicate that there may be a difference in thunderburst locations around this neighborhood. Which two of the five ponds should be monitored?
(b) Determine the most conservative estimate of pond performance (i.e., find $A \cup B$ ).

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04:23

Problem 3

Methane biofilters can be used to oxidize methane using biological activities. It has become necessary to compare performance of two test columns, A and B. The methane outflow level at the surface, in nondimensional units of $X=\{50,100,150,200\}$, was detected and is tabulated below against the respective methane inflow into each test column. The following fuzzy sets represent the test columns:

$$
\mathrm{A}=\left\{\frac{0.15}{50}+\frac{0.25}{100}+\frac{0.5}{150}+\frac{0.7}{200}\right\} \quad \mathrm{B}=\left\{\frac{0.2}{50}+\frac{0.3}{100}+\frac{0.6}{150}+\frac{0.65}{200}\right\}
$$

Calculate the union, intersection, and the difference for the test columns.

Lottie Adams
Lottie Adams
Numerade Educator
06:05

Problem 4

Given a set of measurements of the magnetic field near the surface of a person's head, we want to locate the electrical activity in the person's brain that would give rise to the measured magnetic field. This is called the inverse problem, and it has no unique solution. One approach is to model the electrical activity as dipoles and attempt to find one to four dipoles that would produce a magnetic field closely resembling the measured field. For this problem we will model the procedure a neuroscientist would use in attempting to fit a measured magnetic field using either one or two dipoles. The scientist uses a reduced chi-square statistic to determine how good the fit is. If $\mathrm{R}=1.0$, the fit is exact. If $\mathrm{R} \geq 3$, the fit is bad. Also a two-dipole model must have a lower R than a one-dipole model to give the same amount of confidence in the model. The range of R will be taken as $\mathrm{R}=\{1.0,1.5,2.0,2.5,3.0\}$ and we define the following fuzzy sets for $\mathrm{D}_1=$ the one-dipole model and $\mathrm{D}_2=$ the two-dipole model:

$$
\begin{aligned}
& \mathrm{D}_1=\left\{\frac{1}{1.0}+\frac{0.75}{1.5}+\frac{0.3}{2.0}+\frac{0.15}{2.5}+\frac{0}{3.0}\right\} \\
& \mathrm{D}_2=\left\{\frac{1}{1.0}+\frac{0.6}{1.5}+\frac{0.2}{2.0}+\frac{0.1}{2.5}+\frac{0}{3.0}\right\}
\end{aligned}
$$

For these two fuzzy sets, find the following:
(a) $\mathrm{D}_1 \cup \mathrm{D}_2$
(b) $\underline{D_1} \cap D_2$
(c) $\overline{\bar{D}_1}$
(d) $\overline{\mathrm{D}_2}$
(e) $D_1 \mid D_2$
(f) $\overline{\mathrm{D}_1 \cup \mathrm{D}_2}$

Michaela Flitsch
Michaela Flitsch
Numerade Educator
08:23

Problem 5

In determining corporate profitability, many construction companies must make decisions based upon the particular client's spending habits, such as the amount the client spends and their capacity for spending. Many of these attributes are fuzzy. A client which spends a "large amount" is considered to be "profitable" to the construction company. A "large" amount of spending is a fuzzy variable, as is a "profitable" return. These two fuzzy sets should have some overlap, but they should not be defined on an identical range.

$$
\begin{aligned}
& \underset{\mathrm{A}}{ }=\{\text { ["large' spenders }\} \\
& \underset{\sim}{\mathrm{B}}=\text { ["profitable" clients }]
\end{aligned}
$$

For the two fuzzy sets shown in Fig. P2.5, find the following properties graphically:
(a) $\underset{\sim}{\mathrm{A}} \cup \mathrm{B}$ : all clients deemed profitable or who are large spenders.
(b) $A \cap B$ : all clients deemed profitable and large spenders.
(c) $\overline{\mathrm{A}}$ and $\overline{\mathrm{B}}$ : those clients (i) deemed not profitable, and (ii) deemed not large spenders (separately).
(d) $\overline{\mathrm{A}} \mid \overline{\mathrm{B}}$ : entities deemed profitable clients, but not large spenders.
(e) $\overline{\mathrm{A} \cup \mathrm{B}}=\overline{\mathrm{A}} \cap \overline{\mathrm{B}}$ (De Morgan's principle).

Julian Wong
Julian Wong
Numerade Educator

Problem 6

Suppose you are a soils engineer. You wish to track the movement of soil particles under strain in an experimental apparatus that allows viewing of the soil motion. You are building pattem recognition software to allow a computer to monitor and detect the motions. However, there are two difficulties in "teaching" your software to view the motion: (1) the tracked particle can be occluded by another particle; (2) your segmentation algorithm can be inadequate. One way to handle the occlusion is to assume that the area of the occluded particle is smaller than the area of the unoccluded particle. Therefore, when the area is changing you know that the particle is occluded. However, the segmentation algorithm also makes the area of the particle shrink if the edge detection scheme in the algorithm cannot do a good job because of poor illumination in the experimental apparatus. In other words, the area of the particle becomes small as a result of either occlusion or bad segmentation. You define two fuzzy sets on a universe of nondimensional particle areas, $\mathrm{X}=[0,1,2,3,4]$ : A is a fuzzy set whose elements belong to the occlusion, and $\underset{\sim}{B}$ is a fuzzy set whose elements belong to inadequate segmentation. Let

$$
\begin{aligned}
& A=\left\{\frac{0.1}{0}+\frac{0.4}{1}+\frac{1}{2}+\frac{0.3}{3}+\frac{0.2}{4}\right\} \\
& B=\left\{\frac{0.2}{0}+\frac{0.5}{1}+\frac{1}{2}+\frac{0.4}{3}+\frac{0.1}{4}\right\}
\end{aligned}
$$
Find the following:
(a) $\mathrm{A} \cup \mathrm{B}$
(b) $A \cap B$
(c) $\bar{A}$
(d) $\overline{\mathrm{B}}$
(e) $\overline{\mathrm{A} \cap \mathrm{B}}$
(f) $\overline{\mathrm{A} \cup \mathrm{B}}$

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Problem 7

You are asked to select an implementation technology for a numerical processor. Computation throughput is directly related to clock speed. Assume that all implementations will be in the same family (e.g., CMOS). You are considering whether the design should be implemented using medium-scale integration (MSI) with discrete parts, field-programmable array parts (FPGA), or multichip modules (MCM). Define the universe of potential clock frequencies as $\mathrm{X}=\{1,10,20,40,80,100\} \mathrm{MHz}$; and define MSI, FPGA, and MCM as fuzzy sets of clock frequencies that should be implemented in each of these technologies, where the following table defines their membership values:
Representing the three sets as $\mathrm{MSI}=\mathrm{M}, \mathrm{FPGA}=\mathrm{F}$, and $\mathrm{MCM}=\mathrm{C}$, find the following:
(a) $\mathrm{M} \cup \underset{\sim}{\mathrm{F}}$
(b) $\mathrm{M} \cap \mathrm{F}$
(c) $\bar{M}$
(d) $\overline{\mathrm{F}}$
(e) $\mathrm{C} \cap \overline{\mathrm{F}}$
(f) $\bar{M} \cap \underset{C}{C}$

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Problem 8

We want to compare two sensors based upon their detection levels and gain settings. For a universe of discourse of gain settings, $X=\{0,20,40,60,80,100\}$, the sensor detection levels for the monitoring of a standard item provides typical membership functions to represent the detection levels for each of the sensors; these are given below in standard discrete form:

$$
\begin{aligned}
& S_1=\left\{\frac{0}{0}+\frac{0.5}{20}+\frac{0.65}{40}+\frac{0.85}{60}+\frac{1.0}{80}+\frac{1.0}{100}\right\} \\
& S_2=\left\{\frac{0}{0}+\frac{0.45}{20}+\frac{0.6}{40}+\frac{0.8}{60}+\frac{0.95}{80}+\frac{1.0}{100}\right\}
\end{aligned}
$$

Find the following membership functions using standard fuzzy operations:
(a) $\mu_{\mathrm{S}_1 \cup \mathrm{S}_2}(x)$
(b) $\mu_{\mathrm{s}_1 \cap \mathrm{S}_2}(x)$
(c) $\mu_{51}^{-(x)}$
(d) $\mu_{S_2}(x)$
(e) $\mu \sum_{s_1 \leq \mathbb{S}_1}(x)$
(f) $\mu_{\mathrm{s}_1 \cap \mathrm{s}_1}(x)$

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Problem 9

For flight simulator data the determination of certain changes in operating conditions of the aircraft is made on the basis of hard breakpoints in the Mach region. Let us define a fuzzy set to represent the condition of "near" a Mach number of 0.74 . Further, define a second fuzzy set to represent the condition of "in the region of " a Mach number of 0.74 . In typical simulation data a Mach number of 0.74 is a hard breakpoint.

$$
\begin{aligned}
& \mathrm{A}=\text { near Mach } 0.74=\left\{\frac{0}{0.730}+\frac{0.8}{0.735}+\frac{1}{0.740}+\frac{0.6}{0.745}+\frac{0}{0.750}\right\} \\
& B=\text { in the region of Mach } 0.74=\left\{\frac{0}{0.730}+\frac{0.4}{0.735}+\frac{0.8}{0.740}+\frac{1}{0.745}+\frac{0.6}{0.750}\right\}
\end{aligned}
$$

For these two fuzzy sets find the following:
(a) $\mathrm{A} \cup \mathrm{B}$
(b) $\mathrm{A} \cap \mathrm{B}$
(c) $\stackrel{\sim}{\mathrm{A}}$
(d) $\mathrm{A} \mid \mathrm{B}$
(e) $\overline{\mathrm{A} \cup \mathrm{B}}$
(f) $\overline{\mathrm{A} \cap \mathrm{B}}$

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Problem 10

A system component is tested on a drop table in the time domain, $t$, to shock loads of haversine pulses of various acceleration amplitudes, $\ddot{x}$, as shown in Fig. P2.10a. After the test the component is evaluated for damage. Define two fuzzy sets, "Passed" $=\underset{\sim}{\mathrm{P}}$ and "Failed" $=\mathrm{F}$. Of course, failed and passed are fuzzy notions, since failure to the component might be some partial level between the extremes of pass and fail. These sets are defined on a linear scale of accelerations, $|\vec{x}|$, which is the magnitude of the input pulse (see Fig. P2.10b). We define the following set operations:
(a) $\underset{\sim}{\mathrm{F}} \cup \underset{\mathrm{P}}{\mathrm{P}}=\{|\ddot{x}|\}$ : the universe of input shock level results.
(b) $\mathrm{E} \cap \mathrm{P}$ : the portion of the universe where the component could both fail and pass.
(c) $\overline{\mathrm{F}}$ : portion of universe that definitely passed.
(d) $\overline{\mathrm{P}}$ : portion of universe that definitely failed.
(e) $\mathrm{F} \mid \mathrm{P}$ : the portion of the failed set that definitely failed.

Define suitable membership functions for the two fuzzy sets $\underset{\sim}{F}$ and $\underset{\sim}{P}$ and determine the operations just described.

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Problem 11

Suppose an engineer is addressing a problem in the power control of a mobile cellular telephone transmitting to its base station. Let MP be the medium-power fuzzy set, and HP be the high-power set. Let the universe of discourse be comprised of discrete units of $\mathrm{dB} \cdot \mathrm{m}$, i.e., $X=\{0,1,2, \ldots, 10\}$. The membership functions for these two fuzzy sets are shown in Fig. P2.11. For these two fuzzy sets, demonstrate union, intersection, complement, and the difference.

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Problem 12

Consider a local area network (LAN) of interconnected workstations that communicate using Ethernet protocols at a maximum rate of $10 \mathrm{Mbit} / \mathrm{s}$. Traffic rates on the network can be expressed as the peak value of the total bandwidth (BW) used; and the two fuzzy variables, "Quiet" and "Congested," can be used to describe the perceived loading of the LAN. If the discrete universal set $X=\{0,1,2,5,7,9,10\}$ represents bandwidth usage, in Mbit/s, then the membership functions of the fuzzy sets Quiet $Q$ and Congested $\underset{\sim}{C}$ are as shown in Fig. P2.12.
For these two fuzzy sets, graphically determine the union, intersection, complement of each, difference $\mathrm{Q} \mid \mathrm{C}$, and both De Morgan's principles.

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Problem 13

An engineer is asked to develop a glass break detector/discriminator for use with residential alarm systems. The detector should be able to distinguish between the breaking of a pane of a glass (a window) and a drinking glass. From analysis it has been determined that the sound of a shattering window pane contains most of its energy at frequencies centered about 4 kHz whereas the sound of a shattering drinking glass contains most of its energy at frequencies centered about 8 kHz . The spectra of the two shattering sounds overlap. The membership functions for the window pane and the glass are given as $\mu_{\mathrm{A}}(x)$ and $\mu_{\mathrm{B}}(x)$, respectively. Illustrate the basic operations of union, intersection, complement, and difference for the following membership functions:

$$
\begin{aligned}
x & =0,1, \ldots, 10 \quad \sigma=2 \quad \mu_{\mathrm{A}}=4 \quad \mu_{\mathrm{B}}=8 \mu_{\mathrm{A}}(x)=\exp \left[\frac{-\left(x-\mu_{\mathrm{A}}\right)^2}{2 \sigma^2}\right] \\
\mu_{\mathrm{B}}(x) & =\exp \left[\frac{-\left(x-\mu_{\mathrm{B}}\right)^2}{2 \sigma^2}\right]
\end{aligned}
$$

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Problem 14

Samples of a new microprocessor IC chip are to be sent to several customers for beta testing. The chips are sorted to meet certain maximum electrical characteristics, say frequency and temperature rating, so that the "best" chips are distributed to preferred customer 1. Suppose that each sample chip is screened and all chips are found to have a maximum operating frequency in the range $7-15 \mathrm{MHz}$ at $20^{\circ} \mathrm{C}$. Also, the maximum operating temperature range $\left(20^{\circ} \mathrm{C} \pm \Delta T\right)$ at 8 MHz is determined. Suppose there are eight sample chips with the following electrical characteristics:
The following fuzzy sets are defined:

$$
\begin{aligned}
& \underset{\sim}{\mathrm{A}}=\text { set of "fast" chips }=\text { chips with } f_{\max } \geq 12 \mathrm{MHz} \\
& =\left\{\frac{0}{1}+\frac{0}{2}+\frac{0}{3}+\frac{0}{4}+\frac{0.2}{5}+\frac{0.6}{6}+\frac{1}{7}+\frac{1}{8}\right\} \\
& \mathrm{B}=\text { set of "slow" chips }=\text { chips with } f_{\max } \geq 8 \mathrm{MHz} \\
& =\left\{\frac{0.1}{1}+\frac{0.5}{2}+\frac{1}{3}+\frac{1}{4}+\frac{1}{5}+\frac{1}{6}+\frac{1}{7}+\frac{1}{8}\right\} \\
& \mathrm{C}=\text { set of "cold" chips }=\text { chips with } \Delta T_{\max } \geq 10^{\circ} \mathrm{C} \\
& =\left\{\frac{0}{1}+\frac{0}{2}+\frac{1}{3}+\frac{1}{4}+\frac{1}{5}+\frac{1}{6}+\frac{1}{7}+\frac{1}{8}\right\} \\
& \mathrm{D}=\text { set of "hot" chips }=\text { chips with } \Delta T_{\max } \geq 50^{\circ} \mathrm{C} \\
& =\left\{\frac{0}{1}+\frac{0}{2}+\frac{0}{3}+\frac{0.5}{4}+\frac{0.1}{5}+\frac{1}{6}+\frac{0.5}{7}+\frac{1}{8}\right\}
\end{aligned}
$$

It is seen that the units for operating frequencies and temperatures are different; hence, the associated fuzzy sets could be considered from different universes and operations on combinations of them would involve the Cartesian product. However, both sets of universes have been transformed to a different universe, simply the universe of countable integers from 1 to 8 . Based on a single universe, use these four fuzzy sets to illustrate various set operations. For example, the following operations relate the sets of "fast" and "hot" chips:
(a) $\mathrm{A} \cup \mathrm{D}$
(b) $A \cap D$
(c) $\bar{A}$
(d) $\mathrm{A} \mid \mathrm{D}$
(e) $\overline{\mathrm{A} \cup \mathrm{D}}$
(f) $\overline{\mathrm{A} \cap \mathrm{D}}$

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