PHYS 2341 Portland Community College Measurements in Physics Experiment Measurements in Physics lab.please the attachments for more details about the lab.

PHYS 2341 Portland Community College Measurements in Physics Experiment Measurements in Physics lab.please the attachments for more details about the lab. I attached all what you need for completing the lab. I still have other 5 labs that I will work with you on, if you could do this one. Thanks Mathematics for computing
ICT101_Tri1 2020_Test 3
Test3
Weight
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The Lecture topics
Due date
Total Question
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29/05/2020 starts Friday 8:00 a.m
Chapters 9 and 10.1
30/5/2020 8:00 a.m
5
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First Name:
Family Name:
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Question1
4 Marks
In a competition between players X and Y, the first player to win three games in
a row or a total of four games wins. How many ways can the competition be
played if X wins the first game and Y wins the second and third games? (Draw a
tree.)
Answer here:
Question2
4 Marks
Evaluate the following quantities.
a. P(6, 6)
b. P(6, 3)
c. P(6, 1)
Solution:
Question 3
4 Marks
a. How many integers are there from 1000 through 9999?
b. How many odd integers are there from 1000 through 9999?
c. How many integers from 1000 through 9999 have distinct digits?
d. How many odd integers from 1000 through 9999 have distinct digits?
e. What is the probability that a randomly chosen
four-digit integer has distinct digits? has distinct
digits and is odd?
Answer here:
Question 4
4 Marks
An urn contains two blue balls (denoted B1 and B2) and three white balls (denoted
W1, W2, and W3). One ball is drawn, its color is recorded, and it is replaced in
the urn. Then another ball is drawn
and its color is recorded.
a. Let B1W2 denote the outcome that the first ball drawn is B1 and the second
ball drawn is W2.
Because the first ball is replaced before the second ball is drawn, the outcomes
of the experiment are equally likely. List all 25 possible
outcomes of the experiment.
b. Consider the event that the first ball that is drawn is blue. List all outcomes in
the event. What is the probability of the event?
c. Consider the event that only white balls are drawn. List all outcomes in the
event. What is the probability of the event?
Solution here:
Question5
4 Marks
In the graph below, determine whether the following walks are trails, paths,
closed walks, circuits, simple circuits, or just walks.
a. v1e2v2e3v3e4v4e5v2e2v1e1v0
b. v2v3v4v5v2
c. v4v2v3v4v5v2v4
d. v2v1v5v2v3v4v2
e. v0v5v2v3v4v2v1
f. v5v4v2v1
Solution:
ICT101_ Math For Computing
THEORY OF GRAPHS AND TREES
Theory of Graph I
Week 11 Lecture
Reference:
Chapter 10 of prescribed book: Susanna S. Epp (2020).Discrete
Mathematics with Applications, 5th Edition, Cengage
CHAPTER 10
THEORY OF GRAPHS AND
TREES
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10.1
Trails, Paths, and Circuits
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Trails, Paths, and Circuits
The subject of graph theory began in the year 1736 when
the great mathematician Leonhard Euler published a paper
giving the solution to the following puzzle:
The town of Königsberg in Prussia (now Kaliningrad in
Russia) was built at a point where two branches of the
Pregel River came together. It consisted of an island and
some land along the river banks.
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Trails, Paths, and Circuits
These were connected by seven bridges as shown in
Figure 10.1.1.
The Seven Bridges of Königsberg
Figure 10.1.1
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Trails, Paths, and Circuits
The question is this: Is it possible for a person to take a
walk around town, starting and ending at the same location
and crossing each of the seven bridges exactly once?
To solve this puzzle, Euler translated it into a graph theory
problem. He noticed that all points of a given land mass
can be identified with each other since a person can travel
from any one point to any other point of the same land
mass without crossing a bridge.
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Trails, Paths, and Circuits
Thus for the purpose of solving the puzzle, the map of
Königsberg can be identified with the graph shown in
Figure 10.1.2, in which the vertices A, B, C, and D
represent land masses and the seven edges represent the
seven bridges.
Graph Version of Königsberg Map
Figure 10.1.2
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Trails, Paths, and Circuits
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Trails, Paths, and Circuits
For ease of reference, these definitions are summarized in
the following table:
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Example 10.1.1 – Notation for Walks
a. In the graph below, the notation e1e2e4e3 refers
unambiguously to the following walk: v1e1v2e2v3e4v3e3v2.
On the other hand, the notation e1 is ambiguous if used
by itself to refer to a walk. It could mean either v1e1v2 or
v2e1v1.
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Example 10.1.1 – Notation for Walks
continued
b. In the graph of part (a), the notation v2v3 is ambiguous if
used to refer to a walk. It could mean v2e2v3 or v2e3v3.
On the other hand, in the graph below, the notation
v1v2v2v3 refers unambiguously to the walk v1e1v2e2v2e3v3.
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Example 10.1.2 – Walks, Trails, Paths, and Circuits
In the graph below, determine which of the following walks
are trails, paths, circuits, or simple circuits.
a. v1e1v2e3v3e4v3e5v4
c. v2v3v4v5v3v6v2
e. v1e1v2e1v1
b. e1e3e5e5e6
d. v2v3v4v5v6v2
f . v1
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Example 10.1.2 – Solution
a. This walk has a repeated vertex but does not have a
repeated edge, so it is a trail from v1 to v4 but not a path.
a. v1e1v2e3v3e4v3e5v4
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Example 10.1.2 – Solution
b. This is just a walk from v1 to v5. It is not a trail because it
has a repeated edge.
c. This walk starts and ends at v2, contains at least one
edge, and does not have a repeated edge, so it is a
circuit. Since the vertex v3 is repeated in the middle, it is
not a simple circuit.
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Example 10.1.2 – Solution
continued
d. This walk starts and ends at v2, contains at least one
edge, does not have a repeated edge, and does not
have a repeated vertex. Thus it is a simple circuit.
e. This is just a closed walk starting and ending at v1. It is
not a circuit because edge e1 is repeated.
f. The first vertex of this walk is the same as its last vertex,
but it does not contain an edge, and so it is not a circuit.
It is a closed walk from v1 to v1. (It is also a trail from v1
to v1.)
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Subgraphs
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Subgraphs
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Example 10.1.3 – Subgraphs
List all subgraphs of the graph G with vertex set {v1, v2} and
edge set {e1, e2, e3}, where the endpoints of e1 are v1 and
v2, the endpoints of e2 are v1 and v2, and e3 is a loop at v1.
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Example 10.1.3 – Solution
G can be drawn as shown below.
There are 11 subgraphs of G, which can be grouped
according to those that do not have any edges, those that
have one edge, those that have two edges, and those that
have three edges.
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Example 10.1.3 – Solution
continued
The 11 subgraphs are shown in Figure 10.1.3.
Figure 10.1.3
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Connectedness
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Connectedness
It is easy to understand the concept of connectedness on
an intuitive level. Roughly speaking, a graph is connected if
it is possible to travel from any vertex to any other vertex
along a sequence of adjacent edges of the graph.
The formal definition of connectedness is stated in terms of
walks.
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Connectedness
If you take the negation of this definition, you will see that a
graph G is not connected if, and only if, there exist two
vertices of G that are not connected by any walk.
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Example 10.1.4 – Connected and Disconnected Graphs
Which of the following graphs are connected?
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Example 10.1.4 – Solution
The graph represented in (a) is connected, whereas those
of (b) and (c) are not. To understand why (c) is not
connected, recall that in a drawing of a graph, two edges
may cross at a point that is not a vertex. Thus the graph in
(c) can be redrawn as follows:
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Connectedness
Some useful facts relating circuits and connectedness are
collected in the following lemma.
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Connectedness
A connected component of a graph is a connected
subgraph of largest possible size.
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Example 10.1.5 – Connected Components
Find all connected components of the following graph G.
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Example 10.1.5 – Solution
G has three connected components: H1, H2, and H3 with
vertex sets V1, V2, and V3 and edge sets E1, E2, and E3,
where
V1 = {v1, v2, v3},
V2 = {v4},
E1 = {e1, e2},
E2 =
V3 = {v5, v6, v7, v8},
E3 = {e3, e4, e5}.
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Euler Circuits_ we will discuss it
next week
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ICT101_ Math For Computing
Theory of Graph I
Week 8 Lecture_ Make up class
6/5/2020
Reference:
Chapter 1,9 of prescribed book: Susanna S. Epp (2020).Discrete
Mathematics with Applications, 5th Edition, Cengage
CHAPTER 9
COUNTING AND PROBABILITY
CHAPTER 1
SPEAKING MATHEMATICALLY
Copyright © Cengage Learning. All rights reserved.
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1.4
The Language of Graphs
Copyright © Cengage Learning. All rights reserved.
4
The Language of Graphs
Imagine an organization that wants to set up teams of three
to work on some projects.
In order to maximize the number of people on each team
who had previous experience working together
successfully, the director asked the members to provide
names of their previous partners.
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The Language of Graphs
This information is displayed below both in a table and in a
diagram.
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The Language of Graphs
From the diagram, it is easy to see that Bev, Cai, and Flo
are a group of three previous partners, and so it would be
reasonable for them to form one of these teams. The
drawing below shows the result when these three names
are removed from the diagram.
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The Language of Graphs
This drawing shows that placing Hal on the same team as
Ed would leave Gia and Ira on a team where they would
not have a previous partner.
However, if Hal is placed on a team with Gia and Ira, then
the remaining team would consist of Ana, Dan, and Ed, and
everyone on both teams would be working with a previous
partner.
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The Language of Graphs
Drawings such as these are illustrations of a structure
known as a graph. The dots are called vertices (plural of
vertex) and the line segments joining vertices are called
edges.
As you can see from the first drawing, it is possible for two
edges to cross at a point that is not a vertex. Note also that
the type of graph described here is quite different from the
“graph of an equation” or the “graph of a function.”
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The Language of Graphs
In general, a graph consists of a set of vertices and a set of
edges connecting various pairs of vertices. The edges may
be straight or curved and should either connect one vertex
to another or a vertex to itself, as shown below.
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The Language of Graphs
In this drawing, the vertices are labeled with v’s and the
edges with e’s. When an edge connects a vertex to itself
(as e5 does), it is called a loop.
When two edges connect the same pair of vertices (as e2
and e3 do), they are said to be parallel. It is quite possible
for a vertex to be unconnected by an edge to any other
vertex in the graph (as v5 is), and in that case the vertex is
said to be isolated.
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The Language of Graphs
The formal definition of a graph follows.
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Example 1.4.1 – Terminology
Consider the following graph:
a. Write the vertex set and the edge set, and give a table
showing the edge-endpoint function.
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Example 1.4.1 – Terminology
continued
b. Find all edges that are incident on v1, all vertices that are
adjacent to ν1, all edges that are adjacent to e1, all loops,
all parallel edges, all vertices that are adjacent to
themselves, and all isolated vertices.
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Example 1.4.1 – Solution
a. vertex set = {v1, v2, v3, v4, v5, v6}
edge set = {e1, e2, e3, e4, e5, e6, e7}
edge-endpoint function:
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Example 1.4.1 – Solution
continued
b. e1, e2, and e3 are incident on v1.
v2 and v3 are adjacent to v1.
e2, e3, and e4 are adjacent to e1.
e6 and e7 are loops.
e2 and e3 are parallel.
v5 and v6 are adjacent to themselves.
v4 is an isolated vertex.
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Example 1.4.2 – Drawing More Than One Picture for a Graph
Consider the graph specified as follows:
vertex set = {v1, v2, v3, v4}
edge set = {e1, e2, e3, e4}
edge-endpoint function:
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Example 1.4.2 – Drawing More Than One Picture for a Graph
continued
Both drawings (a) and (b) shown below are pictorial
representations of this graph.
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Example 1.4.3 – Labeling Drawings to Show They Represent the Same Graph
Consider the two drawings shown in Figure 1.4.1. Label
vertices and edges in such a way that both drawings
represent the same graph.
Figure 1.4.1
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Example 1.4.3 – Solution
Imagine putting one end of a piece of string at the top
vertex of Figure 1.4.1(a) (call this vertex v1), then laying the
string to the next adjacent vertex on the lower right (call this
vertex v2), then laying it to the next adjacent vertex on the
upper left (v3), and so forth, returning finally to the top
vertex v1. Call the first edge e1, the second e2, and so forth,
as shown below.
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Example 1.4.3 – Solution
continued
Now imagine picking up the piece of string, together with its
labels, and repositioning it as follows:
21
Example 1.4.3 – Solution
continued
This is the same as Figure 1.4.1(b), so both drawings
represent the graph with vertex set {v1, v2, v3, v4, v5}, edge set
{e1, e2, e3, e4, e5}, and edge-endpoint function as follows:
22
Examples of Graphs
23
Example 1.4.4 – Using a Graph to Represent a Network
Telephone, electric power, gas pipeline, and air transport
systems can all be represented by graphs, as can
computer networks—from small local area networks to the
global Internet system that connects millions of computers
worldwide.
Questions that arise in the design of such systems involve
choosing connecting edges to minimize cost, optimize a
certain type of service, and so forth.
24
Example 1.4.4 – Using a Graph to Represent a Network
continued
A typical network, called a hub-and-spoke model, is shown
below.
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Examples of Graphs
A directed graph is like an (undirected) graph except that
each edge is associated with an ordered pair of vertices
rather than a set of vertices. Thus each edge of a directed
graph can be drawn as an arrow going from the first vertex
to the second vertex of the ordered pair.
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Example 1.4.6 – Using a Graph to Represent Knowledge
In many applications of artificial intelligence, a knowledge
base of information is collected and represented inside a
computer. Because of the way the knowledge is represented
and because of the properties that govern the artificial
intelligence program, the computer is not limited to retrieving
data in the same form as it was entered; it can also derive
new facts from the knowledge base by using certain built-in
rules of inference. For example, from the knowledge that the
Los Angeles Times is a big-city daily and that a big-city daily
contains national news, an artificial intelligence program
could infer that the Los Angeles Times contains national
news.
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Example 1.4.6 – Using a Graph to Represent Knowledge
continued
The directed graph shown in Figure 1.4.2 is a pictorial
representation for a simplified knowledge base about
periodical publications.
Figure 1.4.2
28
Example 1.4.6 – Using a Graph to Represent Knowledge
continued
According to this knowledge base, what paper finish does
the New York Times use?
Meaning of Finish:
complete the manufacture or decoration of (a
material, object, or place) by giving it an
attractive surface appearance.
29
Example 1.4.6 – Solution
The arrow going from New York Times to big-city daily
(labeled “instanceof”) shows that the New York Times is a
big-city daily. The arrow going from big-city daily to
newspaper (labeled “is-a”) shows that a big-city daily is a
newspaper.
The arrow going from newspaper to matte (labeled “paperfinish”) indicates that the paper finish on a newspaper is
matte. Hence it can be inferred that the paper finish on the
New York Times is matte.
30
Examples of Graphs
31
Example 1.4.8 – Degree of a Vertex
Find the degree of each vertex of the graph G shown
below.
32
Example 1.4.8 – Solution
deg(v1) = 0 since no edge is incident on v1 (v1 is isolated).
deg(v2) = 2 since both e1 and e2 are incident on v2.
deg(v3) = 4 since e1 and e2 are incident on v3 and the loop
e3 is also incident on v3 (and contributes 2 to the
degree of v3).
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Example 1.4.9 – Using a Graph to Color a Map
Imagine that the diagram shown below is a map with
countries labeled A–J. Show that you can color the map so
that no two adjacent countries have the same color.
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Example 1.4.9 – Solution
Notice that coloring the map does not depend on the sizes
or shapes of the countries, but only on which countries are
adjacent to which. So, to figure out a coloring, you can
draw a graph, as shown below, where vertices represent
countries and where edges are drawn between pairs of
vertices that represent adjacent countries.
35
Example 1.4.9 – Solution
continued
Coloring the vertices of the graph will translate to coloring
the countries on the map.
As you assign colors to vertices, a relatively efficient
strategy is, at each stage, to focus on an uncolored vertex
that has maximum degree, in other words that is connected
to a maximum number of other uncolored vertices.
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Example 1.4.9 – Solution
continued
If there is more than one such vertex, it does not matter
which you choose because there are often several
acceptable colorings for a given graph.
37
Example 1.4.9 – Solution
continued
For this graph, both C and H have maximum degree so you
can choose one, say, C, and color it, say, blue.
Now since A, F, I, and J are not connected to C, some of
them may also be colored blue, and, because J is
connected to a maximum number of others, you could start
by coloring it blue.
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Example 1.4.9 – Solution
continued
Then F is the only remaining vertex not connected to either C
or J, so you can also color F blue. The drawing below shows
the graph with vertices C, J, and F colored blue.
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Example 1.4.9 – Solution
continued
Since the vertices adjacent to C, J, and F cannot be colored
blue, you can simplify the job of choosing additional colors by
removing C, J, and F and the edges connecting them to
adjacent vertices. The result is shown in Figure 1.4.4a.
Figure 1.4.4(a)
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Example 1.4.9 – Solution
continued
In the simplified graph again choose a vertex that has a
maximum degree, namely H, and give it a second color, say,
gray.
Since A, D, and E are not connected to H, some of them may
also be colored gray, and, because E is connected to a
maximum number of these vertices, you could start by
coloring E gray.
41
Example 1.4.9 – Solution
continued
Then A is not connected to E, and so you can also color A
gray. This is shown in Figure 1.4.4b.
Figure 1.4.4(b)
42
Example 1.4.9 – Solution
continued
The drawing below shows the original graph with vertices
C, J, and F colored blue, vertices H, A, and E, colored gray,
and the remaining vertices colored black. You can check
that no two adjacent vertices have the same color.
43
Example 1.4.9 – Solution
continued
Translating the graph coloring back to the original map
gives the following picture in which no two adjacent
countries have the same color.
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