Experience at Monta Vista
Debbie Frazier at Monta Vista High School has incorporated a segment on logic in two different
courses: (1) Digital Innovation and Design (DID) (2) AP Computer Science and Principles. We
discuss these courses in a bit more detail next. Both of these courses have been offered at
Monta Vista starting Fall 2016.
Digital Design and Innovation
In the Digital Innovation and Design (DID) course, the students will explore the potential of
technology to solve modern human problems by applying design principles used in art and
business. They will design, field test, and evaluate their own projects that use technology to
solve local problems. The course will examine the "magic" of how computers and the Internet
work, and how logic helps digital devices communicate and understand one another. The
course will strengthen students’ critical thinking and logic skills as they evaluate the effective
use of technology and weigh the legal and ethical questions that may occur with technological
solutions. Students will be exposed to a range of professions that use digital technology.
Students who complete this course successfully will be prepared for Java Programming.
The DID course is suitable for 9th grade students who are currently in Algebra 1 (as
opposed to Geometry and above.) It provides a path towards Java programming class (and, for
students who wish to continue, AP Computer Science A or AP Computer Science Principles). The
course is aimed to both accelerate the learning of students who need to increase their logical
reasoning and algorithmic thinking skills while also expanding their knowledge of the impact of
technology and computer science on society and our culture. The course design was inspired
by the need that some students did not know what coding or programming was, and a few felt
that programming led to “bad things” in the world. To address these needs, the course
includes a balance of high interest, fun, hands-on activities alongside complex learning related
to key critical thinking skills that students need to be successful in Programming Java and
beyond. Also important to us was to keep the course broad enough that if the student decided
that she did not wish to continue with programming, she could use the skills from the class in
other classes.
Logic is introduced in the DID course as Unit 5 titled as "Using Logic to Get what you
Need". The goal here is to address the following common core standard:
MP.3: Construct viable arguments and critique the reasoning of others.
("Mathematically proficient students are also able to compare the effectiveness of two
plausible arguments, distinguish correct logic or reasoning from that which is flawed,
and—if there is a flaw in an argument— explain what it is.")
In addition, this unit also addresses the following standards defined by Computer Science
Teachers Association:
CPP.L3A-12: Describe how mathematical and statistical functions, sets, and logic are
used in computation.
CT.L2-14: Examine connections between elements of mathematics and computer science
including binary numbers, logic, sets and functions.
Key Learning outcomes from this Unit are as follows:
 Identify logical ambiguity in an English language description, thereby identifying
limitations to programmed information
 Use a matrix, Venn Diagram, and/or truth table to organizing attributes/relationships to
formalize a logical process and reach a conclusion
 Use logic to write queries to answer practical questions
o Translate English or mathematical statements into a formal logical syntax using
AND, OR, and NOT
 Possible Project: Describe how to do a targeted search using boolean
logic (e.g. with a search engine like Google); subsequently how logic
reduces our amount of effort or improves efficiency
 Identifying attributes needed to be collected to best help a person for a
purpose. This could be exemplified with a digital product and how they
are used with parameters and conditions to make decisions (e.g.
entertainment: Netflix Movie Suggestion, Amazon Shopping Suggestion)
or in a non-digital context for handling real-world issues: moving to a new
home (education, income, safety stats), placing a new grocery store
(Whole Foods’ parameters versus Safeway’s parameters); automated
search for legal discovery, e.g. filter emails for ones pertaining to a
particular person, subject, in time interval).
 Students would identify flaws related to the attributes collected or how
they are collected that can bias conclusions.
 Formulate a set of if-then rules describing a pre-defined set of behaviors or the behavior
of a real thing we are trying to model
 Write a testing plan for rules including edge cases, execute the test, and subsequently
report success, failure, “bugs”
 Logic of debugging: Describe observed “symptom”, list possible sources of observed
behavior, how to systematically identify the source, and possible solutions to reduce
them, based on logic. This could be within the context of a system (real or digital) that
was provided to the student.
o (e.g. (1) If I come to school and no one is here… (a) It could be Saturday, (b)
Everyone could be in an assembly, (c)...; (2) Your mom calls and says her phone
isn’t working. What questions do you ask to deduce the problem? Or, e.g. direct
comparison with medical diagnosis)
This Unit will use first three chapters and the exercises from the Logic textbook as the course
material. In addition, the students will work on projects that involve constructing Boolean
queries for a search engine, and Excel Macros to query a publicly available dataset.
The DID course has been approved as a “g” course by UC. The course is currently being
offered in the Fremont Union High School District by Akane Akaban --- a teacher who we
trained during Summer of 2016. She has, however, not gone through the module on logic at the
time of preparation of this report.
AP CS Principles Course
Initial Design Philosophy
AP CS Principles is a computer science course teaching about design, application, and
limitations to digital technology. Students are expected to be users of technology, and either
have (a) a background in programming, or (b) a background in design/art. Students will work
collaboratively to explore and create technology. Enrolled students are expected to complete
two projects submitted to the College Board and will take the College Board APCSP exam in
May 2017.
AP CS Principles is a second or third year Computer Science course. Students who are
enrolled in AP CS Principles typically take Java Programming in the 9th grade and AP CS
Principles in the 10th, 11th, or 12th grade. AP CS Principles is not open to freshmen, as it is a
college-level course.
The material on logic will be introduced as part of Unit 0 and Unit 1 of this course. Unit
0 is titled as Setting the Stage: Science Principles, Defining Need, Abstraction, and
Interconnectivity of Business, Law, and Product Evolution. Unit 1 is titled as Programming
Principles.
In Unit 0, logic will be covered in the context of the following topics: Students will
review basic hardware components including logic gates in transistors, in chips, in cards or
other memory devices. Bits of information extrapolated to data more meaningful to the
student will pave the way to reviewing data types and structures.
Students will read, write, and analyze logic notation in conjunction with logic gate
diagrams and truth tables. Limitations in logic with respect to natural language and required
sets of information will be introduced in simplified models, including identifying problems as
decidable or undecidable. Students will use several diagramming techniques to explore
algorithm correctness, efficiency, and complexity.
As part of homework, the students will work with truth tables and compound conditions using
logical notation and plain English. Examples using natural language will also be used, for
students to identify limitations in logic, for instance identifying limited required data
sets/variables for satisfiability using logic, and when algorithms are undecidable or decidable.
Use of symbolism and diagramming steps of logical conclusion with truth tables and trees, will
enable students to determine efficiency (count of conclusions), correctness, and complexity of
algorithms. Using digger.com or powerset.com students will test and understand limitations to
natural language interpretation. The students will be expose to captcha (Completely Automated
Public Turing test to tell Computers and Humans Apart). Chapters 1-3 of the Logic textbook will
be used for this Unit. The use of above activities in this unit will address the following essential
knowledge required by the AP Computer Science Standard:
EK 2.2.3.F: A logic gate is a hardware abstraction that is modeled by a Boolean function.
EK 2.2.3.G: A chip is an abstraction composed of low-level components and circuits that
perform a specific function.
EK 2.2.3.H: A hardware component can be low level like a transistor or high level like a
video card.
EK 2.2.3.I: Hardware is built using multiple levels of abstractions, such as transistors,
logic gates, chips, memory, motherboards, special purpose cards, and storage devices.
In Unit 1, logic will be covered in the context of the following topics: Students will use
logical notation and application of logic to natural language, classification, and filtering.
As an exercise for this segment, the students will work on creating a taxonomic knowledge base
for cells (prokaryotic, eukaryotic) with mitochondria and chloroplasts as interesting exceptions.
Students will encode the activity at the electron transport chain generally (to work
for prokaryotes, mitochondria, and chloroplast), and add AP CS Principles Course Syllabus 20167 specialization as appropriate. Students will be given a series of questions at several levels in
Bloom’s Taxonomy to answer, and have to trace the paths used to answer each. This is also
used as an example for exploring artificial intelligence and processing of natural language, as
the questions will be in plain English and the answers are meant to be insightful for questions at
the higher levels of Bloom’s Taxonomy. This exercise will be done on paper only.
In a second homework, the students will incorporate logic into control theory (say, for a
thermostat, then later for something more complex). Students will draw out a system and
define variables, procedures (including parameters and return types). Focus will be on distinct
conditions and evaluating compound conditions with truth tables, using extreme cases.
Students will be exposed to very complex cases and encouraged to get as far as they can with
formal proof in these cases. They will explore impacts to efficiency based on one to several
CPUs working in parallel. Chapters 1-4 of the Logic textbook will be used for this segment. The
use of above activities in this unit will address the following essential knowledge required by
the AP Computer Science Standard:
EK 4.1.1.A: Sequencing, selection, and iteration are building blocks of algorithms.
EK 4.1.1.B: Sequencing is the application of each step of an algorithm in the order in
which the statements are given.
EK 4.1.1.C: Selection uses a Boolean condition to determine which of two parts of an
algorithm is used.
EK 4.2.4.A: Determining an algorithm’s efficiency is done by reasoning formally or
mathematically about the algorithm.
EK 4.2.4.B: Empirical analysis of an algorithm is done by implementing the algorithm and
running it on different inputs.
EK 4.2.4.C: The correctness of an algorithm is determined by reasoning formally or
mathematically about the algorithm, not by testing an implementation of the algorithm.
EXCLUSION STATEMENT: Formally proving program correctness is beyond the scope of
this course and the AP Exam.
EK 5.5.1.E: Logical concepts and Boolean algebra are fundamental to programming.
EK 5.5.1.F: Compound expressions using and, or, and not are part of most programming
languages.
EK 5.5.1.G: Intuitive and formal reasoning about program components using Boolean
concepts helps in developing correct programs.
nally.
Posthoc feedback
Debbie implemented her plan of incorporating Logic into her offering of the Advanced
Placement Course on computer science principles starting Fall’2016. We describe here some of
her experiences.
APCSP is a design course in which students have to distill problems, programs, and
existing products down to foundational elements that are connected by functions and relations
to produce solutions or useful output. A small amount of logic is required in the course (as
defined by the College Board). She embraced the opportunity to teach more as a means to
emphasize how elegantly systems can be described, particularly when it comes to defining
something for a computer (which ultimately must process decisions in terms of 0s and 1s or
false/trues). APCSP is a programming-language agnostic course, so she saw using logic
formalism as a way to be concrete in how something could be communicated to a machine or
another person. There are numerous parallels between formalism and other logical means of
communication and how we communicate in programming and natural languages.
Furthermore, the frequent struggles we have to translate something in natural language,
address exceptions in a system/world, or communicate something emotional or aesthetic into a
logical formalism helps to emphasize that there will always be some things we cannot encode.
Debbie followed our Introduction to Logic textbook in the order presented. Each
chapter is not very long, which is helpful. To help students synthesize the material into a
learning set that is easier to come back to, rather than have them take notes as they read
(students are notoriously bad at summarizing or paraphrasing new material), she provided
reading questions (included in the appendix). Just prior to turning these in, she discussed any
confusion on this “first pass” through the text, then students submit their work for credit
(partly completion, partly accuracy). She then proceeded to incorporate the lesson’s
vocabulary and lessons into applied contexts. This very much built up over time, so by the time
students collaborated on projects using topics of interest, computer science, design, and other
key concepts from the class, they were revisiting logic concepts and vocabulary from several
chapters. When students designed software or analyzed existing digital tools, they regularly
assessed them for 6 things: decidability, soundness, completeness, correctness, efficiency, and
polymorphism. Students got into the habit of explaining how the tool/software addressed each
of these 6 things, and proofs, propositional resolution, flow charts, and/or truth tables regularly
were included, to substantiate their claims.
Students found the first chapter interesting, as it is tied to some interesting issues in
semantics - them misinterpreting headlines in newspapers, for instance. They also enjoyed
using matrices to solve questions about sets/systems. As the course got into formalism,
students were much more hesitant, as writing included Greek symbols and they struggled to
differentiate implication (less intuitive) and entailment (more intuitive). Students had trouble
starting proofs without premises, but most enjoyed how quickly one could arrive at a solution
with propositional resolution. Chapter 6 has been very popular, probably because it brings
things “full circle” with what the course tried to use logic for in the very beginning of the year to describe systems.
In teaching programming in a different course, there is a clear distinction between
students who do and do not truly own logical expressions and the associated process of
applying them or thinking of all versions of them, when trying to search, filter, or sort large sets
of data. After applying logic so regularly and relatively thoroughly in this class (APCSP) that
students are considerably more aware of their process in defining algorithms to search, sort,
and filter large sets of data, and they are evaluating their logical expressions far more
thoroughly in the APCSP class.
Debbie will teach logic in her Fall’2017 AP CS Principles class. As a refinement, she will
overtly review more frequently some concepts that she found were difficult for students, but
retain the pacing she used this year (Chapters 1-5 first semester, Chapters 6-8 2nd semester).
Appendix
The appendix here contains several supplemental materials that Debbie designed for her class.
These were real challenges raised amongst professionals studying Artificial Intelligence and issues
with Natural Language in 2013. We’ll just unpack the problems and solutions in class/for homework.
Do the best you can do. DO NOT take more than 1 hour to do this homework, please. We will discuss
and get farther together. 
Drug-drug interaction mining
https://www.cs.york.ac.uk/semeval-2013/task9.html
It is important for doctors/medical staff, pharmacists, researchers, and people taking drugs to know
about drug-drug interactions. This includes supplement-drug interactions.
For instance: “Vitamin D helps the body produce melatonin.” In this case, the interaction is that
Vitamin D is a precursor for melatonin being made by the body.
“Acetazolamide reduces urinary excretion of quinidine.” Acetazolamide prevents quinidine
from leaving the body (so it is active in the body for longer).
There are more examples here: https://www.cs.york.ac.uk/semeval2013/task9/data/uploads/task-9.2-ddi-extraction.pdf or you can pick up a pharmaceutical (e.g. bottle of
Advil) and read warnings.
1. How do we determine if there is an interaction that is negative, positive, or neutral for the
person?
2. How do we determine if there is an issue with fatality or addiction?
3. How do we combine information from sources of different types and formats, to know what
is well researched (and group research together) so we know what is NOT yet well
understood?
4. How do we look at descriptions of interactions and create semantic meaning when
acronyms, common names, brand / generic names, and chemical names are all used?
Twitter analysis for sentiment
https://www.cs.york.ac.uk/semeval-2013/task2.html
Twitters are very short text entries. They can contain slang, misspellings, shorthand, hashtags, and
URLs.
1. How do we look at a twitter or text message and determine if it is positive, negative, or neutral?
2. How do we look at a twitter or text message and determine if it is hate speech/cyberbullying?
3. How do we look at twitter/text messages to amass info that is all about the same topic; How do
we look at a twitter or text message set and determine what is popular news to promote/push
to others, from a marketing standpoint? (Use several metrics)
4. How do we look at a twitter or text message and create semantic meaning when slang,
shorthand, misspelling, hashtags, or URLs are used in replacement of or in addition to more
“literal” text?
Each person will work on one numbered question above for one of the cases, defining the problem and
coming up with some ideas on paper.
Drug-1
Drug-2
Drug-3
Drug-4
Twitter-1
Twitter-2
Twitter-3
Twitter-4
Focus on:
•
What data would be stored in files.
•
Data types you would use. One would be words. You can think about meta-data like
length of words or some other feature of the words as well.
•
Additional data that would be tracked, for semantic meaning. This is often much
harder, as there will be exceptions or special cases.
•
Define some decision-making rules: if ----(something about input)--- then ---(assignment of semantic value)---. Multi-decision points are fine.
•
Try to formalize the rules using logic notation.
•
Come up with at least three things to test on that are real for that context.
•
Where are issues with testing/semantics/rules? Are these high-stakes or low-stakes
based on the context of your project (Twitter or Medicine)?
SemEval case #3: Noun compounds and synonyms
Name
____________________________________ Period ___
Noun compounds are nouns made up of several words. You are to consider how to make a digital tool
that would determine if some text is a synonym of a noun compound. For instance, banana bread could
also be bread with a key ingredient banana*. (* Yes, this is a very awkward example of a synonym –
Mrs. Frazier doesn’t want to do all of your good thinking for you!)
Just make your tool work with banana bread synonyms.
List some words or phrases to look for that would indicate you have a synonym or not. Put at least
three of each in each semantic box below.
Words for a synonym of a noun compound
Words for a non-synonym of a noun compound
Consider some additional data you could use to confirm that you are working with a synonym of banana
bread. (*No, there is no thesaurus of banana bread, so think more broadly than that.) Describe where
you would get this data or how that data works (explain briefly). Write at least one unique piece in each
space below:
-
Confirming synonym
-
Confirming non-synonym
Now let’s put your two pieces of information together. Let’s do this formally (using logical notation) and
informally (in English). Use the following test examples. Make sure the words here are not in your
answers above – Mrs. F wants to make sure you can come up with your own ideas too.
Let w be true when the word/words in your text are in your synonym semantic group. Let s be true when
the other data is in your confirming synonym semantic group. Let c be true when the overall synonym
checks out as a synonym for banana bread. The first example has been started for you. You need to
decide about what s checks for and fill that in.
Statement
Bread with the key ingredient banana
w
1
s
c
1
Add your own test
examples as well!
Write a compound statement using w, s, and c to summarize what is being tested in your cases. Use
symbols for AND, OR, NOT, IMPLIES, or BICONDITIONAL IMPLICATION.
Chapter 1 Logic Reading: http://logic.stanford.edu/intrologic/notes/chapter_01.html (Introductory Logic Textbook
by Michael Genesereth, a professor at Stanford University). The goal of this sheet is to help you summarize
important content and make additional connections (note bold & underlined terms below).
1.
“Logic is increasingly being used by c_________________ - to prove mathematical theorems, to validate
engineering d_________________, to diagnose failures, to encode and analyze laws and regulations and
b________________ rules.”
2.
Notice in the first diagram in section 1.2 that it is labeled as a state of the sorority system. A system is a
set of data that are related, often with some kind of behaviors or actions. A state is how the system can
be described, based on values for each piece of data.
Data here is comprised of names such as ___________________ and status of liking / being liked.
Relationships are based on liking or not liking other people.
Sadly, “Nobody ______________ herself.”
The system is defined in a matrix showing the data and relationships.
3.
A set of premises l_________________ e_____________ a conclusion if and only if every world that
satisfies the premises also satisfies the conclusion.”
4.
Logical r________________________________ (application of reasoning rules that leads to logical
conclusions) leads to logical p_________________.
5.
A Rules of Inference: Copy the first general rule of inference (it has been started for you; specific versions
were used with Hondas and slithy toves, but write the general one here):
All x are y.
___________ _____ are z.
Therefore, _______ ______ are ______.
6.
Deduction: “What distinguishes a correct pattern from one that is incorrect is that it ________
_______________________lead to correct conclusions … logically entailed by the premises.”
7.
Induction: “reasoning from the p_________________ to the g__________________.”
8.
Abduction: reasoning from e_________________ to possible c______________.
9.
Reasoning by analogy: “infer a conclusion based on _________________________ of two situations.”
10. Classify the type of reasoning. Tell if you think the logic is faulty or not.
Example
The more books students read, the more they
learn. Comics are like books. Therefore, the
Type
Faulty or
Not?
If faulty, what is wrong?
more comics read, the more a student will
learn.
I have eaten 30 times at Chipotle. I have never
had food poisoning from eating at Chipotle.
Therefore, Chipotle will not make me sick ever.
If there is no food, dinner will not be made. If
there is no cook, dinner will not be made.
There is a cook. Dinner has not been made.
Therefore, there is no food.
All leaves are photosynthetic. All
photosynthetic things have pigments.
Therefore, all leaves have pigments.
11. Translation of Natural Language to logic (known as Formalization) can be difficult because languages can
be c____________ and am_________________.
12. Two contrasting pictures help to define the difference in meaning between the girl-and-telescope
examples. Choose another example from Section 1.3 and draw two rough pictures making clear two
distinct meanings for the same line.
13. In the reading, the following is written and explained in English. Complete the English translation.
p⇒q
p ______________________ q
m ∧ r⇒p ∨ q
(m ___________ r) _________________ (p ___________ q)
14. Resolution: the only process needed to s_____________ any problem.
15. Dr. Genesereth has a fantastic list of fields in which automation could assist humans in completing work.
We’ll use many of these this year! Fill in the blanks.
Engineers: “write specifications for their products and to encode their d_____________… simulate
designs… diagnose failures and to develop t_________________ programs.”
Database Systems: “…compute new tables, to detect problems, and to optimize qu__________________.”
Logical Spreadsheets: “… in scheduling applications, we might have timing constraints or restrictions on
who can reserve which rooms. In the domain of travel reservations, we might have c____________________ on
adults and infants. In academic program sheets, we might have constraints on how many courses of varying types
that students must take.”
Law and Business: “… encode r_____________________________ and business r_________________,
and automated reasoning techniques can be used to analyze such regulations for inconsistency and overlap.”
16. Propositional Logic: uses a______, o____, and n_____.
17. Relational Logic: uses propositional logic and adds constants, var_________________, and
quan___________________________.
Complete exercises 1-4. You will find these at the bottom of the reading or there are interactive versions
accessed through the exercises link (http://logic.stanford.edu/intrologic/exercises/exercises.html). You do not
need to show any work – if you complete the interactive versions, you will get real-time, hands-on feedback.
Note that we will have a quiz using these same kinds of questions on Monday.
Chapter 2 Logic Reading Questions (reading only – we’ll do online problems on another day)
1. Propositional logic concerns propositions. “A propositions is a possible condition of the world
that is either ____________ or ______________.”
2. Propositional logic has two sentence types: ___________ sentences (simple facts, also called
proposition/logical constants) and ______________ sentences (logical relationships between the
simple sentences).
3. Complete the table below of the five kinds of compound sentences. Recall that Chapter 1
defined what the symbols mean. To help you, use this list of words: and, or, not, implies, p, and
q.
Type of sentence
Example from reading
What example means
(¬p)
Conjunction
Disjunction
Implication
(p ⇔ q)
4. In the appropriate example above (there is one), label the antecedent and the consequent.
5. In math class, you learned about operator precedence, and perhaps learned to memorize it with
PEMDAS, which Mrs. Frazier likes to write as PE(MD)(AS) because addition (A) and subtraction(S)
have equal priority, handled left-to-right. The logic reading has its own operator precedence.
Write the precedence order:
6. If there are two operators next to an operand (a term that is not an operator), then the operator
with _________________ (higher or lower) precedence is applied. If the two operators have
equal precedence, then the one on the _________________ (left or right) is applied first.
Find an example from the list to exemplify this:
7. The reading describes meta-level notation. For instance, in case i, p is set to the value true. This
is written by p = ____. Note that meta-knowledge is knowledge that is dependent on a specific
case or example.
8. For negation, a true sentence becomes ______________ (true or false).
9. For conjunction, ________________ (one or both) sentences need to be true for the
conjunction to be true.
10. For disjunction, ________________ (one or both) sentences need to be true for the disjunction
to be true.
11. “The truth value of an implication is false if and only if its antecedent
is _____________ and its consequent is __________; otherwise, the truth value
is true.”
12. The truth table for implication is shown at right. What do you notice about when the
two sentences are false? The implication is __________ (true or false). Note that this
may make more sense if you read the logical statements: “false implies false” “true
implies true.”
φ
ψ
1
1
0
0
1
0
1
0
13. “A biconditional is true if and only if the truth values of its constituents agree, i.e.
they are either both _____________ or both _______________.”
φ
ψ
14. If you are paying close attention, you’ll be surprised by the result from the third row
of values in the two truth tables shown (the second one is for biconditonals). What do
you notice?
1
1
0
0
1
0
1
0
15. What does it mean for a sentence to be satisfied?
16. What is the opposite of a sentence being satisfied? The sentence is _f__________________.
17. “for a propositional language with n proposition constants, there are n columns in the truth table
and _____ rows.” For this reason, a truth table is not ideal for testing satisfiability for large sets
of proposition constants.
18. “If” and “Only if” should help you differentiate between propositional logic sentences. Write
the two examples presented in the reading, using the formal notation with c, f, and p. Next to
each, write the corresponding English sentence.
19. The logical statements you wrote above are both true. Write them as a single biconditional with
c, p, f, and appropriate symbols.
φ⇒
ψ
1
0
1
1
φ⇔
ψ
1
0
0
1
Logic and Gate Practice within Circuits (Classwork)
From Chapter 2: “At a given point in time, a node in a circuit can be either ______ or _____.” The output
of an and gate is on if and only if both of its inputs are ____. The value of an or node is on if and only if
at least one of its inputs is ____. The output of an xor gate is on if and only if its inputs
__________________ with each other.”
Intro to Logic, Chapter 3
Name __________________________________ Period ___
You’ll notice that the interface has changed a bit. Still find the readings at
http://intrologic.stanford.edu/lessons/lessons.html. Some new symbols for you:
=| means “entails”
|# means “doesn’t imply”
U means union for a set (e.g. group a OR b OR both)
1. Validity vs. Contingency vs. Unsatifiability
a. Explain why Validity and Unsatisfiability are opposites of eachother.
b. Explain why Validity and Unsatisfiability are relatively useless or uninteresting, while
Contingency is useful or interesting.
c. Satisfiability only applies to which of these three traits (list two)?
d. Falsifiability only applies to which of these three traits (list two)?
2. How is a truth table used to prove logical equivalence?
3. Logical Entailment: A person has to be 18 and a registered voter to vote. Don’t introduce
additional variables other than the age and state of registration for the problems below.
a) Describe a situation in which a person is 19 and is entailed to vote.
b) Describe a situation in which a person is 19 and not entailed to vote.
c) True or false (circle one): A person’s age of at least 18 entails an ability to vote.
d) True or false (circle one): A person’s ability to vote entails that the person is at least 18.
4. Logical Entailment, continued: A person has to be a Monta Vista regional resident or a Senior
who has lived in the Monta Vista region in the past to attend MVHS. (Yes, Seniors can commute
in from places like Fremont, and still go to school here.) Do not introduce any other variables
other than the living region to the problems below.
a) True of false (circle one): Living in MV region entails attending MVHS.
b) True of false (circle one): Attending MVHS entails living in the MV region.
c) At least one of the statements (a) and/or (b) above are false. Explain.
5. Review what you hopefully learned by doing problems 3 and 4. What is the conclusion about
logical entailment with a compound statement with AND, compared to logical entailment with a
compound statement with OR?
6. Section 3.4, note (slide) #7 presents the idea of checking satisfiability for a whole set of
premises. If we narrow our judgement of logical entailment for a compound statement with
AND to asking if the whole set of {p,q} entails it, is there logical entailment? (circle one) Yes / No
7. For statements to be logically consistent, ________ statements w and z must be true. What
goes in the blank?
All
At least one
All except one
None
8. For statements to be NOT logically consistent, __________ statements s and t must be true.
What goes in the blanks?
All
At lease one
All except one
None
9. Give an example of statements that are logically consistent but are not logically equivalent and
do not show logical entailment.
10. A biconditional gh makes the two statements g =>h and h=>g true because of the
E__________________________ T______________________.
11. Deduction Theorem states that j entails k if and only if _________________ is valid. It also
states that a set of statements {a, b, c} entails a compound statement a ___ b ___ c if and only if
the compound statement is valid.
12. Unsatisfiability Theorem states that a set m entails a statement n if and only if m union with (not
n) is _______________________________. * See top of reading questions for explanation of a
union.
13. Consistency Theorem states that a is consistent with b if and only if ______________is
_______________________________. The set c is consistent with a finite set of sentences only
if c1 __ c2 ___ c3… is ____________________.
14. Metalevel has to do with prop_______________________ and rel_____________________.
Intro to Logic Chapter 4 Questions
Name _________________________________ Period _
1. Implication Elimination and Implication Creation are opposites. Write the proof (all three lines
including the bar) for Implication Elimination, then label it “Implication Elimination.” To its right,
do the same thing for “Implication Creation.” Label premises and point an arrow to them, then
label and point arrows at conclusions.
2. Do this same thing for Implication Distribution.
3. “A rule applies to a set of sentences if and only if there is ____________________ of the rule in
which ____ of the premises are in the set.”
4. “In using rules of inference, it is important to remember that they apply only to
__________________________________, not to components of sentences.” For instance, if I
know p => (r AND s) and I know p, then I can conclude ______ but not ___ or ___. Fill in the
blanks with r, s, or (r AND s).
5. Mrs. Frazier loves linear proofs. When she taught geometry, she required these. Students were
required to indicate when a premise was being used by writing “given” or referencing the figure,
when an axiom was being used by naming it (often the name of a postulate, property, or
theorem), and the last line of the proof was always the conclusion. For instance, see the proof
below. Mrs. Frazier always required her students to add reference numbers so it was clear what
prior steps’ knowledge (sort of meta-premises) were being used in a new step.
Reference #s
1.
2.
3.
4.
5.
Figure
(1)
-(2, 3)
(4)
Label the geometric proof’s parts: premise, axioms, and conclusion. Put Δ next to the premises
(you are defining a variable). Put w next to the conclusion (again, you are defining a variable).
6. In the geometric proof shown, angle APC’s congruence to angle BPD is provable from the figure
shown using what rules? List them all (for the example shown). These will be called R.
7. Write the formal representation of the English sentence in #6, using R, Δ, |-, and w.
8. Write the shorter version of what you wrote in #7, assuming R is clear from context: _________
9. Structured proofs can include a__________________________ in a subproof.
10. However, using the answer from #9 in structured proofs can be dangerous, if we try to use this
to make a case for proving something in the superproof. Here’s a layman’s example that works
the same way as the first sample in the reading that has a fault. Explain why all is OK at the
subproof level, but not at the superproof level.
Given: If it’s Sunday, I should get dressed up. If I’m dressed up, I should wear a bib when I eat.
Prove: I should wear a bib when I eat.
Statement
1. If it’s Sunday, I should get dressed up.
2. If I’m dressed up, I should wear a bib when I eat.
(subproof) 3. It’s Sunday.
(subproof)4. I should get dressed up.
(subproof)5. I should wear a bib when I eat.
6. If it’s Sunday, I should wear a bib.
7. I should wear a bib.
Reason
Premise (Given)
Premise (Given)
Assumption
Implication Elimination
Implication Elimination
Implication Introduction
Implication Elimination
Reference number
---1, 3
2, 4
3, 5
3, 6
11. Explain what this structured rule of inference, Implication Introduction, says in English.
φ |- ψ
φ⇒ψ
12. Fitch is a p__________ system using ____ rules of inference. (last blank is a number)
13. The Fitch rules presented help you understand why a rule would be called a – Introduction or a –
Elimination. Put either name in each blank:
________________ : building up a compound statement
________________ : breaking down a compound statement
14. In And Elimination, a conclusion is one of the conjucts in the compound AND statement. In Or
Elimination, the conclusion is not a disjunct, but a statement implied. Why?
15. Negative Elimination allows us to work with double n______________________.
16. “a proof system is sound if and only if ______ provable conclusion is logically
_________________________.”
17. “a proof system is complete if and only if ______ logical conclusion is _________________.”
Chapter 5 Logic Reading Questions
In Chapter 4, we began looking at proofs. We continue grouping sets of information in order to make
conclusions here, and a lot of you will like this approach better, as it creates something simple very
quickly.
1. { } is called an empty set. (You may have used this in math as well.) This is also referred to as a
__________________ disjunction. It is used to show that you have a
contra______________________. (Read on until an example later, to get that final answer.)
2. Let’s rewrite some rules:
a) Implication:
A => B means the same as (NOT ___) OR _____
To make sense of this, recall that in implication, the following are true:
False => False, True => ________, _________ => _________
And there is only one statement that is not true in implication:
True => _______
So, if A => B is true, then A could be False OR B could be True.
b. Negation:
¬(φ ∧ ψ) means the same as ____________________
You process this problem using something similar to Distributive Property; NOT AND is
the same as OR. Draw arrows showing how the NOT gets distributed to each symbol in the case
above.
Use the same arrow notation to show how Distribution should be used to find the expanded
version of ¬(φ ∨ ψ)
c. Distribution:
φ ∨ (ψ ∧ χ) means the same as _______________________
Use the same arrow notation to show how the expansion works.
You will notice that the method you used above to expand only applies when we are looking at
complementary signs/symbols. Explain how things are different when the symbols are the same.
Use one of the last four examples under “Distribution” in the reading to help you explain.
d. Operators:
You’ll notice clauses are defined differently, depending on operators. So for clause {a,b,c}, this
can be expanded to mean a ___ b ___ c. What does {a}, {b} mean? It means a ___ b.
3. Propositional Resolution or Resolution Principle should seem familiar based on what you learned
of solving systems in algebra. You’ll see that if you have two clauses that complement in some
way, for instance, one containing p and another containing NOT p, when you combine the two,
you find p _________________________________________________.
4. However, solving of systems isn’t a perfect analogy. If one clause has q and the other has q, you
don’t get 2q when you combine them. Instead you get what? (mention q) _____________
5. You’ll notice that for {p,q} and {NOT p, NOT q} you cannot simply get { }. Recalling {p ,q}
means p OR q, this should make sense. Explain.
Furthermore, there is a rule that should help you rush along, forgetting what {p ,q} means: “Only ______
____________ of literals may be resolved at a time.”
6. The following example is used in the reading to show how to convert with our old way of proof
and our new way using these rules.
p⇒q
p
{¬p, q}
{p}
q
{q}
Convert the following example, drawing a new proof using clauses on the right. Remember to
combine your clauses to get a result at the end.
p⇒q
NOT q
NOT p
7. “A resolution derivation of a conclusion from a set of premises is a ____________ sequence of
clauses terminating in the conclusion in which each clause is either a _____________ or the
_________________ of applying the Resolution Principle to earlier members of the sequence.”
8. “If a set of clauses is unsatisfiable, then there is guaranteed to be a resolution derivation of the
empty clause from this set.”
9. Reference numbers are used with the resolution derivation, like shown below. Explain what 1,2
means here, by showing what pieces are used together and what is concluded, completing the
mini resolution derivation at right (the bar is drawn for you).
1. {p, q} Premise
2. {p, ¬q} Premise
3. {¬p, q} Premise
4. {¬p, ¬q} Premise
5. {p}
1, 2
6. {¬p}
3, 4
7. {}
5, 6
___________
10. True or False?
You don’t need premises to prove something.
________________
11. “One of the best features of Propositional Resolution is that it much more ________________
than the other proof methods we have seen. There is no need to test each and every case
(instance).”
12. “Moreover… Propositional Resolution can be used in a proof procedure that always terminates
without losing completeness.” What does completeness mean again? You can refer to this
textbook’s glossary or go back earlier in the book to read about it.
13. Exercises
Exercise 5.1: Convert the following sentences to clausal form.
(a) p ∧ q ⇒ r ∨ s
(b) p ∨ q ⇒ r ∨ s
(c) ¬(p ∨ q ∨ r)
(d) ¬(p ∧ q ∧ r)
(e) p ∧ q ⇔ r
Exercise 5.2: What are the results of applying Propositional Resolution to the following pairs of clauses.
(a) {p, q, ¬r} and {r, s}
(b) {p, q, r} and {r, ¬s, ¬t}
(c) {q, ¬q} and {q, ¬q}
(d) {¬p, q, r} and {p, ¬q, ¬r}
Exercise 5.3: Use Propositional Resolution to show that the clauses {p, q}, {¬p, r}, {¬p, ¬r}, {p, ¬q} are
not simultaneously satisfiable.
Exercise 5.4: Given the premises (p ⇒ q) and (r ⇒ s), use Propositional Resolution to prove the
conclusion (p ∨ r ⇒ q ∨ s).
Chapter 6 Logic Reading Questions (part 1)
Period ___
Name __________________________________
1.
In chapters 2-5, we focused on Propositional Logic. In Chapter 6, we start working with Relational Logic.
Relational Logic uses v___________________ and q_______________________. We will be using these
to describe objects.
2.
In Relational Logic there are variables and two kinds of constants: o__________ constants and
r_________________ constants. Together with the relationships between these, is formed a vocabulary.
The terms in this vocabulary are the variables and the o___________ constants.
You may want to read ahead to see definitions for vocabulary and term, above, but if not, the answer for
o______________ is the same for both.
3.
Complex expressions can be created by combining relation constants with other arguments. “Each
relation constant has an associated arity, i.e. the ________________ of arguments with which that
_______________________________ can be combined.”
4.
Complete the list below:
Unary means 1 relation constant with ____________ argument(s)
Binary means 1 relation constant with ____________ argument(s)
Ternary means 1 relation constant with ___________ argument(s)
N-ary means 1 relation constant with ____________ argument(s)
5.
In the reading, the sentence q(a,y) is used. This relational sentence describes the relationship
between relation constant q and the terms a and y. Those terms a and y are either variables or object
constants. For the following sentences, complete the table:
w (p, q)
Type of thing
Letter(s) used
Relation Constant
Term (Object constant or variable)
Arity of relation constant (circle one)
Unary
Binary
Ternary
N-ary
Ternary
N-ary
m (v, h, s)
Type of thing
Letter(s) used
Relation Constant
Term (Object constant or variable)
Arity of relation constant (circle one)
6.
Unary
Binary
The following occur in relational logic as well as propositional logic (circle all that apply)
Conjunction
Negation
Biconditional Implication
Disjunction
7.
“Quantified sentences are formed from a quantifier, a __________________, and an embedded
________________. The embedded sentence is called the scope of the quantifier. There are two types of
quantified sentences… u__________________ quantified sentences and e____________________
quantified sentences.”
8.
Complete the table to compare quantified sentences.
Type of quantified sentence
Symbol used in
notation
Used to assert that ___ objects have a
property (fill in the blank with all or
some)
Universal
Existential
9. Ground means a relational expression contains no ______________________.
10. Free and bound are used to describe variables in relational logic. For this example, label the variables as
free or bound:
∃c.a(b,c)
11. What is the difference between an open and closed sentence? Use an example to help you explain.
12. Semantics of relational logic is also known as __________________ semantics.
13. You can write the Herbrand base for any described set of object constants and relation constants. An
example is given using a and b and p and q. Using that as a guide, create the Herbrand base for a
vocabulary with object constants m and v and relation constants s and r, where s has arity 1 and r has
arity 2.
14. “For a set of terms of size b, there are bn distinct n-tuples of object constants; and hence there
are bn ground relational sentences for each n-ary relation constant.” Apply this to your example above.
In question 13, there were ___ terms (that’s “size b”). When arity was 1, n = 1, so there were ___
relational sentences with that arity. When arity was 2, n = 2, so there were ___ relational sentences with
that arity (you’re using “bn”).
15. Now try applying the ideas from #13 and #14 to this example: How many relational sentences will there
be, and show all of them, considering the Herbrand base for a vocabulary with object constants c, s, and p
and relation constants r and x where r has arity 2 and x has arity 3.
16. Truth assignments can be made for each ground relational sentence in a Herbrand base. Once we have
truth assignments, we can check for satisfiability just like we did with propositional logic. Free variables
will need to be replaced by ground terms so that we can determine if a relational sentence is true (1) or
false (0). Because the free variable is truly meant to vary, replacing it with a ground term really only
creates one example or instance of that expression. This is similar to instance creation or instantiation
when we make objects. We talked about this in class with respect to object mapping (when we talked
about metadata and definition of types of people) and with respect to caching (in which instances are
made once and facilitate a website using lots of data grouped to build objects, loads faster). Below left is
an example of your object that could be used in logic – label it “OBJECT”. The two boxes on the right are
derived from this object – label hem “INSTANCES.” Complete the second example of an instance and
complete the sentence by filling in the blank.
∃c.a(b,c)
Sentence:
a(b,c)
a(2,c)
“c is divided by 2 to give a remainder of zero”
a(___,c)
“_____________________________________”
“for some numbers c, c is
divided by b to give a
remainder of zero”
17. Sometimes relational sentences have more than one free variable. “An instance of an expression is an
expression in which ____ free variables have been consistently replaced by ground terms. Consistent
replacement here means that, if one occurrence of a variable is replaced by a ground term, then ___
occurrences of that variable are replaced by the same ground term.”
18. “A universally quantified sentence is true for a truth assignment if and only if ______ instance of the
scope of the quantified sentence is true for that assignment. An existentially quantified sentence is true
for a truth assignment if and only if _____ instance of the scope of the quantified sentence is true for that
assignment.”
19. “A truth assignment satisfies a sentence with free variables if and only if it satisfies ________ _________
of that sentence. A truth assignment satisfies a ____ of sentences if and only if it satisfies __________
sentence in the ____.
20. Read the first paragraph of the section on Evaluation, then revisit your table in question #8. Add a column
and label it “how many instances must be true to make sentence true.” Fill in the cells for that column
with one or all.
For home after this assignment is turned in: Online exercises 6.1 and 6.3
Chapter 6 Logic Reading Questions (Part 2)
Name____________________________ Period ___
21. Consider Sorority World.
a) How would you encode Bess likes Cody AND Dana?
b) How would you encode Abby likes everyone that Cody likes? You are encouraged to write
how this would be paraphrased, as per the similar example in the reading, to make this
easier to understand.
c) How would you encode Nobody likes Dana?
d) Explain how the encoding is different for “everyone likes someone” and “someone likes
everyone.” This example is used in the reading – you are encouraged to show the encoding,
but please offer an explanation in your own words.
22. Consider Blocks World.
a) Blocks World is more interesting than Sorority World because the number of relation
constants is larger. What are the five relation constants? Complete the list: on,
____________________________
b) Draw the Blocks World using blocks (object constants) a, b, c, and d in which the following
are true:
on(b,a)
on(c,b) ¬on(c, a)
¬on(d, c)
¬on(d, b)
¬on(d, a)
table(d) stack(c,b,a)
c) Complete the statements using the relation constant clear and one object constant each,
meeting the requirements for Blocks World in the immediately previous question. There are
two possible answers; show both. clear ( ____ )
clear (____ )
d) Copy the rule in English and write the formalism next to it. This should help you be able to
read/write with logic.
English: “A block satisfies the table relation ________________ it is ___ on some block.”
Formalism: ______________________________
English: “Three blocks satisfy the stack relation if and only if the first is on the second and
the second is on the third.” Formalism: ______________________________
23. Now use what you have learned in the previous question to write formalism for an English
sentence, for the relation same used with numbers: Two numbers satisfy the same relation if and
only if the first is equal to the same number as the second one is equal to.
∀a. ∀b. ∀c. (same (a, b) ______ same (a,c) ____ same (b, c))
24. Consider Modular Arithmetic in a system using modulus of 4 (so remainders can only be 0, 1, 2,
or 3). Complete the formalism in (a) and answer the question in (b).
a) There are some numbers a and c for which they satisfy the next relation, then a > c.
___a. ___ c. ( next (a, c) _____ a > c)
b) There is in fact one set of numbers that would make (a) true. What are they? If you need
help, you’ll find all possible instances of next(a, b) in the reading. a = ___ and c= ____
c) True or false (circle one)? This is satisfiable: ∀x. ∀y. (next(x, y) ⇒ x<y)
25. Match each term to its example, by writing the term before it. You will use some more than
once. Terms are:
Common Quantifier Reversal
Negation Distribution
Existential Distribution
______________________ ∃x.∃y.q(x,y) ⇔ ∃y.∃x.q(x,y))
______________________ ∃y.∀x.q(x,y) ⇒ ∀x.∃y.q(x,y)
______________________ ¬∀x.p(x) ⇔ ∃x.¬p(x)
______________________ ∀x.∀y.q(x,y) ⇔ ∀y.∀x.q(x,y)
______________________ ¬∃x.p(x) ⇔ ∀x.¬p(x)
26. Consider a thermostat (like one on your wall to regulate when the heater turns on and off).
Let a be the temperature of the ambient air. Let b be the ideal temperature (the temperature
the thermostat is set to).
onHeat(a,b) if and only if a< b. offHeat(a,b) if and only if a > b. Write formalism for these two
universal rules.
Now consider a more complex thermostat that can not only heat by turning on a heater, but can
cool by turning on an air conditioner. Complete English rules and logical formalism for onCool
and offCool.
onCool(a,b) if and only if _________. offCool(a, b) if and only if __________.
Formalism:
You can also write about onCool and offCool in terms of onHeat and PART of offHeat. Give it a
try by writing in English and formal logic notation below:
Complete a truth table for the instances below.
State of
a>b a<b a=b onHeat(a,b) offHeat(a,b) onCool(a,b) offCool(a,b)
variables
for
instance
onHeat(a,b)
Ʌ
offCool(a,b)
a=75,
b=70
a= 55,
b=70
a= 70,
b=70
You can stop reading at 6.10; all of our old vocabulary applies to Relational Logic.
Complete 6.4, 6.5, 6.6 online at home after you
turn this in.
Practice Test for AP CS Principles Unit 0 Exam
Hillary Clinton’s staff facilitated her using an unsecure server for email communication. Hillary got in
trouble because some of the emails were supposed to be secure.
1. What need statement is most appropriate for this situation? (circle one)
a. People who did not “grow up” in the information age need assistance understanding data
permanence and risk, because if they don’t, they can jeopardize security and safety.
b. People in high profile positions need to leave technical work up to others, because technical
experts can better keep information secure.
c. Hillary Clinton needs to only access and use classified information at her professional office,
because it is not possible to make home email servers secure.
d. Email servers need to have good levels of security, because espionage and identity theft is a
real problem.
2. The FBI investigated the data that was insecure on Hilary’s server. They used a digital tool to help
look through thousands of emails, replacing a person who would have had to do the same thing.
This digital tool is (circle one):
smart tech.
assistive tech.
Turing tech.
3. The tool would read through emails and look for terms that would identify the contents of the email
as being classified (needing to be secure or secret) or not classified. In each labeled bin below, place
3 words, relationships, or phrases that are associated with that semantic bin. Don’t use “classified,”
“secure,” or “secret” – emails are not so overt. Instead, think about the type of information that
would be in the email.
Classified
Not Classified
4. The FBI tool then reviewed the widely known news at the time of each email to confirm if the
information was classified or not. This input could come from (circle all that apply):
a) Blogs, forums, or editorials commenting on and sharing opinions about political news
b) Reputable news websites (e.g. CNN, New York Times)
c) Facebook, Instagram, or other social media news sites
d) Military briefs amongst high level officials discussing military strategy and news from warzones
5. The tool reads in an email’s date, then uses that to check in the set of data from the source(s) you
picked in #4. Here, the date would be (circle two) ROM
RAM
volatile
involatile
6. Consider the tool and how it works on its most basic abstract level. The macro elements are: (circle
all that apply)
Hilary
Server with Emails
Database from #4
Date
House
7. The tool’s micro elements are: (circle all that apply)
Date
Classified state of text in emails
E-mails
State of security of the server
Matching state of text in news to emails
Rows
A
8. Examples are shown below that could be used to test the tool. If c is true, then there is a semantic
match to classified words/phrases (your answer in #3). If n is true, then there is a match between
the idea in the email and the idea in news (your choice in #4). If ideas in these examples match
what you used in #3, then please add to your ideas in #3 to make sure you meet #3’s
requirements.
Complete the truth table.
Columns 
1
2
3
4
5
Example
c
n
g
c => g
c |- g
Email contents: Come to coffee on Friday.
Strawberry shortcake made by Amir.
News: <No news match about coffee, strawberry,
shortcake, Amir recently prior to this email>
B
Email contents: Chinese building islands, claiming
land in South China Sea.
0
News: China and Philippines dispute over man-made
islands.
C
Email contents: Military scheduling fly-over in Syria to
show US presence.
1
News: <no news match about Military, fly-over, Syria,
US presence recently prior to this email>
D
Email contents: Military drone strike strategy for ISIS
compound housing leader X.
1
News: Drone strike planned on ISIS leaders.
9. When you test your tool with the four examples above, your results match the data in your truth
table. This means that:
a) The problem you are solving is (circle one)
undecidable
decidable
b) Which is satisfiable? Columns (circle all that apply)
4
5
c) Which is falsifiable? Columns (circle all that apply)
4
5
d) Which is contingent? Columns (circle all that apply)
4
5
e) Which is valid? Columns (circle all that apply)
4
5
f) In what rows is c  g true? (circle all that apply)
A
B
C
D
g) Does n logically entail g? (circle one)
Yes
No
h) Is n logically consistent with g? (circle one)
Yes
No
i) Is n logically equivalent with g? (circle one)
Yes
No
10. You use Google search tool to get your news-searching tool to work. This (circle all that apply)
a) Can use site: to specifically look for matching news in your specific news sites.
b) Makes Google’s search tool polymorphic
c) Can take in a specific date to limit search results
d) Uses spiders that query the deep web
11. You design a test of your tool on a random set of Hillary’s emails. This includes what step(s)? (circle
all that apply) Not all steps in the process are necessarily listed.
a) Numbering all emails in a list
b) Randomly generating a number
c) Picking every 10th email in the set of emails
d) Picking one email on each day emails were sent
12. One limitation of your tool is its ability to work with attached files. When the files are text
documents, then the tool works, because it can read words, but when the attachments are graphics
files, including scans (photos) of text, the tool has trouble. This is because (circle all that apply)
a) Algorithms are not good at interpreting graphics
b) Scans of text can include extra marks (like bits of dust) that break up the word (similar to
how CAPTCHA works)
c) Technology is not able to convert color to number values / make decisions about color
d) Technology cannot be used to open attachments
13. IF the tool could work with graphics, it would have to determine the semantic meaning of the
graphics. IF a tool could do this, it could also be used to evaluate copyright infringement. If two
documents (A and B) could be used as evidence for copyright infringement, then (circle all that
apply)
a) A fails a Turing Test, but B passes the Turing Test.
b) Someone cannot tell the difference between A and B.
c) A and B represent the same thing (e.g. A is a drawing of a happy dog, B is a photo of a happy
dog)
d) B is a modification of A, but the meaning for both is the same.
e) B is a modification of A, but the meaning for each is different.
14. E-mail’s impact on the world has paralleled the impact of Burden’s Wheel. How? (circle all that
apply)
a) E-mail began as a tool for a limited few, but now almost everyone uses it.
b) E-mail networks us (even if we live far apart or are in different time zones).
c) E-mail has made very powerful and central our email tools/providers. Billions of people turn
to Google (gmail) or Yahoo (yahoo mail) for instance.
d) E-mail made mail communication cheaper, faster, and easier (than paper mail
communication).
e) E-mail makes life better – we can get help, social assurance, and speak our minds.