Proofs
Daily Number: 22
• 22/7  
• 222=484
• 22 is a pentagonal number
• There are 22 letters in the Hebrew alphabet
Abercrombie’s lawsuit claims that its Hollister Co. stores own the
exclusive trademark rights to use the number "22" on clothing since
Hollister has been using the number from the time the chain opened in
2000 and since the number is well-recognized as representing 1922, the
year the company pretends it was started. Neither Abercrombie & Fitch nor
Hollister Co. ever formally registered the number as a trademark, but the
lawsuit asserts the company gained common law rights to the number
because Hollister Co. has "become widely known and recognized by its
usage of 22."
Abercrombie filed the lawsuit in the federal district court located in
Columbus, Ohio, and is demanding that American Eagle stop using the
number, or anything similar, destroy any clothing or advertising using the
number, and pay Hollister for any profits American Eagle reaped by using
"22." American Eagle has denied the charges, stating in a letter to Hollister
Co. that any use of the number "22" is purely "ornamental." And, in any
event, the "lawsuit is frivolous," according to American Eagle’s General
Counsel, Neil Bulman.
If it isn’t, one could certainly wonder whether the next target will be all
those football jerseys with the "similar" number "33" – for Dallas Cowboys
Hall-of-Famer Tony Dorsett.
Abercrombie’s lawsuit claims that its Hollister Co. stores own the
exclusive trademark rights to use the number "22" on clothing since
Hollister has been using the number from the time the chain opened in
2000 and since the number is well-recognized as representing 1922, the
year the company pretends it was started. Neither Abercrombie & Fitch nor
Hollister Co. ever formally registered the number as a trademark, but the
lawsuit asserts the company gained common law rights to the number
because Hollister Co. has "become widely known and recognized by its
usage of 22."
Abercrombie filed the lawsuit in the federal district court located in
Columbus, Ohio, and is demanding that American Eagle stop using the
number, or anything similar, destroy any clothing or advertising using the
number, and pay Hollister for any profits American Eagle reaped by using
"22." American Eagle has denied the charges, stating in a letter to Hollister
Co. that any use of the number "22" is purely "ornamental." And, in any
event, the "lawsuit is frivolous," according to American Eagle’s General
Counsel, Neil Bulman.
If it isn’t, one could certainly wonder whether the next target will be all
those football jerseys with the "similar" number "33" – for Dallas Cowboys
Hall-of-Famer Tony Dorsett.
• Let C(x,y)= “x is a student in class y”
 xyz((xy)(C(x,z)C(y,z))
More examples
• Let L(x,y)=“x loves y”
 Write “There is somebody who loves nobody besides
himself or herself.”
• Write each of these in symbols. Which of these are
true about this class?
 “There is a student is in this class who is neither a
mathematics major nor a freshmen”
 There is a year of study such that there is a student in
every major in this class in that year of study.
• Suppose the u.d. is the integers. Is the following
true:
 xyz(y/x=z)
Nature & Importance of Proofs
• In mathematics, a proof is:
 a correct (well-reasoned, logically valid) and complete
(clear, detailed) argument that rigorously & undeniably
establishes the truth of a mathematical statement.
• Why must the argument be correct & complete?
 Correctness prevents us from fooling ourselves.
 Completeness allows anyone to verify the result.
• In this course (& throughout mathematics), a very
high standard for correctness and completeness of
proofs is demanded!!
Applications of Proofs
• An exercise in clear communication of logical
arguments in any area of study.
• The fundamental activity of mathematics is the
discovery and elucidation, through proofs, of
interesting new theorems.
• Theorem-proving has applications in program
verification, computer security, automated
reasoning systems, etc.
• Proving a theorem allows us to rely upon on its
correctness even in the most critical scenarios.
Proof Terminology
• Theorem
 A statement that has been proven to be true.
• Axioms, postulates, hypotheses, premises
 Assumptions (often unproven) defining the
structures about which we are reasoning.
• Rules of inference
 Patterns of logically valid deductions from
hypotheses to conclusions.
More Proof Terminology
• Lemma - A minor theorem used as a steppingstone to proving a major theorem.
• Corollary - A minor theorem proved as an easy
consequence of a major theorem.
• Conjecture - A statement whose truth value has
not been proven. (A conjecture may be widely
believed to be true, regardless.)
• Theory – The set of all theorems that can be
proven from a given set of axioms.
Graphical Visualization
A Particular Theory
A proof
The Axioms
of the Theory
Various Theorems
…
Inference Rules - General Form
• An Inference Rule is
 A pattern establishing that if we know that a set
of antecedent statements of certain forms are all
true, then we can validly deduce that a certain
related consequent statement is true.
• antecedent 1
antecedent 2 …
 consequent
“” means “therefore”
Inference Rules & Implications
• Each valid logical inference rule
corresponds to an implication that is a
tautology.
• antecedent 1
Inference rule
antecedent 2 …
 consequent
• Corresponding tautology:
((ante. 1)  (ante. 2)  …)  consequent
Some Inference Rules
•
p
 pq
• pq
p
•
p
q
 pq
Rule of Addition
Rule of Simplification
Rule of Conjunction
p= “I own three bicycles.”
q=“I weigh 225 lbs”
Modus Ponens & Tollens
•
p
pq
q
• q
pq
p
“the mode of
affirming”
Rule of modus ponens
(a.k.a. law of detachment)
Rule of modus tollens
“the mode of denying”
p= “I own three bicycles.”
q=“I weigh 225 lbs”
Syllogism Inference Rules
•
pq
qr
pr
• pq
p
q
Rule of hypothetical
syllogism
Rule of disjunctive
syllogism
p= “I own three bicycles.”
q=“I weigh 225 lbs”
Aristotle
(ca. 384-322 B.C.)
Formal Proofs
• A formal proof of a conclusion C, given
premises p1, p2,…,pn consists of a sequence
of steps, each of which applies some
inference rule to premises or previouslyproven statements (antecedents) to yield a
new true statement (the consequent).
• A proof demonstrates that if the premises
are true, then the conclusion is true.
Formal Proof Example
• Suppose we have the following premises:
“It is not sunny and it is cold.”
“We will swim only if it is sunny.”
“If we do not swim, then we will canoe.”
“If we canoe, then we will be home early.”
• Given these premises, prove the theorem
“We will be home early” using inference rules.
Proof Example cont.
• Let us adopt the following abbreviations:
 sunny = “It is sunny”; cold = “It is cold”;
swim = “We will swim”; canoe = “We will
canoe”; early = “We will be home early”.
• Then, the premises can be written as:
(1) sunny  cold (2) swim  sunny
(3) swim  canoe (4) canoe  early
Proof Example cont.
Step
1. sunny  cold
2. sunny
3. swimsunny
4. swim
5. swimcanoe
6. canoe
7. canoeearly
8. early
Proved by
Premise #1.
Simplification of 1.
Premise #2.
Modus tollens on 2,3.
Premise #3.
Modus ponens on 4,5.
Premise #4.
Modus ponens on 6,7.
Inference Rules for Quantifiers
• x P(x)
P(o)
(substitute any specific object o)
• P(g)
(for g a general element of u.d.)
x P(x)
• x P(x)
P(c)
(substitute a new constant c)
• P(o)
(substitute any extant object o)
x P(x)
Identify the inference rule
Kangaroos live in Australia and are
marsupials
Kangaroos are marsupials
Simplification
p^q
therefore q
Identify the inference rule
It is hotter than 100 degrees today or the
pollution is dangerous
It is below 100 today
 The pollution is dangerous
Disjunctive Syllogism
pq
¬p
therefore q
Identify the inference rule
Linda is an excellent simmer
If Linda is an excellent simmer
then Linda can work as a lifeguard
 Linda can work as a lifeguard
Modus ponens
p
pq
therefore q
Identify the inference rule
Steve will work at a computer company this
summer
 Steve will work at a computer company
this summer or Steve will be a beach bum
Addition
p
therefore p  q
Identify the inference rule
If I work all night on homework then I can
answer all the questions
If I can answer all the questions then I can
understand the material
 If I work all night on homework then I can
understand the material
Hypothetical Syllogism
pq
qr
therefore p  r
What conclusions can be drawn?
• If I work, it is either sunny or partly sunny.
• I worked last Monday or I worked last
Friday.
• It was not sunny on Tuesday.
• It was not partly sunny on Friday.
If I worked on Friday then it was sunny
If I worked on Tuesday then it was partly sunny.
What conclusions can be drawn?
• Every student has an MVNU email account.
• Homer does not have an MVNU email
account.
• Maggie has an MVNU email account
Homer is not a student.
Identify the inference rules
• Each of five roommates, Melissa, Aaron, Ralph,
Veneesha, and Keeshawn, has taken a course in
discrete mathematics.
• Every student who has taken a course in discrete
math can take a course in algorithms.
• Therefore all five roommates can take a course in
algorithms next year.
Identify the inference rules
•
•
•
Each of five roommates, Melissa, Aaron, Ralph, Veneesha, and Keeshawn, has
taken a course in discrete mathematics.
Every student who has taken a course in discrete math can take a course in
algorithms.
Therefore all five roommates can take a course in algorithms next year.
R(x) = “x is a roommate”
D(x) = “x has taken a course in discrete mathematics”
A(x) = “x can take a course in algorithms”
1.  x(R(x)  D(x)) From 1st statement
2.  x(D(x)  A(x)) From 2nd statement
3.  x(R(x)  A(x)) Universal Generalization from (1) & (2)
Using Hypothetical Syllogism
Identify the inference rules
• There is someone in this class who has been
to France.
• Everyone who goes to France visits the
Louvre.
• Therefore someone in this class has visited
the Louvre
Identify the inference rules
•
•
•
There is someone in this class who has been to France.
Everyone who goes to France visits the Louvre.
Therefore someone in this class has visited the Louvre
C(x) = “x is in this class”
F(x) = “x has been to France”
L(x) = “x has visited the Louvre”
1. x(C(x)  F(x)) From 1st statement
2. x(F(x)  L(x)) From 2nd statement
3.  x(L(x))
Existential Generalization from (1) & (2)