Negating Nested Quantifiers
(xy xy  1)
More examples:
𝐶 𝑥, 𝑦 =``student 𝑥 is enrolled in class 𝑦”
1. ¬(∃𝑥 ∀𝑦 𝐶 𝑥, 𝑦 )
2. ∃𝑥 ∃𝑦 ∀𝑧 ( 𝑥 ≠ 𝑦 ∧ 𝐶 𝑥, 𝑧 → 𝐶 𝑦, 𝑧 )
I(x)=“x has an internet connection”
internet”
Domain is students in your class.
C(x,y)=“x and y have chatted over the
1. Someone in your class has an Internet connection but has not
chatted with anyone else in the class.
2. There are two students in the class who between them have
chatted with everyone else in the class.
Section 1.5 – Rules of Inference
• Terms:
– Argument
– Premises
– Conclusion
– Valid
Standard Rules of Inference
(Each is based on a tautology)
Modus Ponens
p
pq
q
Modus Tollens
q
pq
 p
Hypothetical Syllogism
pq
qr
pr
Standard Rules of Inference
(Continued)
Addition
Simplification
Conjunction
p
pq
pq
p
p
q
pq
Standard Rules of Inference (Continued)
Disjunctive Syllogism
pq
p
q
pq
Resolution
p  r
q  r
Examples
Alice is a mathematics major. Therefore, Alice is either a
mathematics major or a computer science major.
If it snows today, the university will close. The university is not
closed today. Therefore, it did not snow today.
If I go swimming, then I will stay in the sun too long. If I stay in
the sun too long, then I will sunburn. Therefore, if I go
swimming, then I will sunburn.
Example
Use rules of inference to show that the hypotheses “Randy works hard,” “If
Randy works hard, then he is a dull boy” and “If Randy is a dull boy, then
he will not get the job” imply the conclusion “Randy will not get the job.”
Rules of Inference for Quantified
Statements
Universal
Instantiation
Universal
Generalization
x P( x)
 P(c) for any fixed value c
P(c) for an arbitrarily chosen value c
 x P( x)
Rules of Inference for Quantified
Statements (Continued)
Existential
Instantiation
Existential
Generalization
x P( x)
 P(c) for some valuec
P(c) for some particularvalue c
 x P( x)
Combining Rules of Inference for
Quantified Statements
Universal Modus
Ponens
Universal Modus
Tollens
x ( P( x)  Q( x))
P (c )
 Q (c )
x ( P( x)  Q( x))
Q(c)
 P(c)
Examples: Drawing Conclusions
“Every computer science major has a personal computer.” “Ralph
does not have a personal computer.” “Ann has a personal
computer.” “Joe is a computer science major.”
Valid Arguments vs Fallacies
• Valid arguments are constructed using…
• A fallacy is a (so-called) argument which is not so
constructed.
– Affirming the conclusion
– Denying the hypothesis
– Begging the question
(( p  q)  q)  p
(( p  q)  p)  q
p
Examples: Valid Argument or Fallacy?
1. All students in this class understand logic. Xavier is a student
in this class. Therefore, Xavier understands logic.
2. Every computer science major takes discrete mathematics.
Natasha is taking discrete mathematics. Therefore, Natasha
is a computer science major.
3. All parrots like fruit. My pet bird is not a parrot. Therefore, my
pet bird does not like fruit.
4. Everyone who eats granola every day is healthy. Linda is not
healthy. Therefore, Linda does not eat granola every day.
Section 1.6 – Introduction to Proofs
Formal Proofs
Definitions:
Proof-
Theorem-
PropositionAxiom or postulate-
Definitions Continued:
Lemma-
Corollary-
Conjecture-
Quantifiers
• Remember that when no quantifier is given, a
universal quantification is assumed.
If xy > 0, then either x and y are both positive or x and y
are both negative
Some basic facts/definitions we’ll need:
• An integer 𝑛 is even if there exists an integer 𝑘 such that 𝑛 = 2𝑘.
• An integer 𝑛 is odd if there exists an integer 𝑘 such that 𝑛 = 2𝑘 +
1.
• An integer 𝑎 is a perfect square if there is an integer 𝑏 such that
𝑎 = 𝑏2 .
• If a and b are integers with 𝑎 ≠ 0, we say that 𝑎 divides 𝑏 if there
is an integer 𝑐 such that 𝑏 = 𝑎𝑐.
• The real number 𝑟 is rational if there exist integers 𝑝 and 𝑞 with
𝑞 ≠ 0 such that 𝑟 = 𝑝/𝑞. A real number that is not rational is
called irrational.
Methods of Proving 𝑝 → 𝑞
(Given arbitrarily complicated compound propositions p and q)
Direct proof: Assume p is true. Show by a direct argument
that q is true.
Task: Prove the statement: “If a person
likes math then he/she is cool.”
Proof:
Example: Prove by a direct argument that if 𝑛 is a perfect square
then 𝑛 is either odd or divisible by 4.
Methods of Proving 𝑝 → 𝑞
(Given arbitrily complicated compound propositions p and q)
Indirect proof: Assume q is false. Show by a direct
argument that p is false.
Task: Prove the statement: “If a person
likes math then he/she is cool.”
Proof:
Example: Prove by an indirect argument that if 𝑚 and 𝑛 are
integers and 𝑚𝑛 is even, then either 𝑚 or 𝑛 must be even.
Proving 𝑝 ↔ 𝑞
1. Show that p→q
2. Show that q→p
Task: Prove the statement: “A person likes
math if and only if he/she is cool.”
Proof:
Proving Multiple Statements Equivalent
Prove these statements are equivalent, where a and b are real numbers: (i) a is less
than b, (ii) the average of a and b is greater than a, and (iii) the average of a and b is
less than b.
Other Types of Proof
• Vacuous proof
• Trivial proof
• Proof by contradiction
Prove that the product of a non-zero rational numbers and an
irrational number is irrational using proof by contradiction.
Mistakes in Proofs
1.
2.
3.
4.
5.
6.
7.
𝑎=𝑏
𝑎2 = 𝑎𝑏
𝑎2 − 𝑏 2 = 𝑎𝑏 − 𝑏 2
𝑎−𝑏 𝑎+𝑏 =𝑏 𝑎−𝑏
𝑎+𝑏 =𝑏
2𝑏 = 𝑏
2=1
Given
Multiply both sides by a
Subtract 𝑏 2 from both sides
Factor
Divide by 𝑎 − 𝑏
Substitute 𝑎 for 𝑏 since 𝑎 = 𝑏
Divide both sides by b
Section 1.7 – Proof Methods and Strategy
• Proof by cases
• Exhaustive Proof
Prove that for any two real numbers 𝑥 and 𝑦, 𝑥 + 𝑦 ≤ 𝑥 + 𝑦 .
Theorems and Quantifiers
• Existence proofs (constructive vs. nonconstructive)
Constructive: Show that there is a positive
integer that can be written as the sum of cubes
of positive integers in two different ways.
Nonconstructive: Prove that there exists two irrational numbers 𝑥
and 𝑦 for which 𝑥 𝑦 is rational.
Uniqueness quantifier and uniqueness proofs
∃! 𝑥 𝑃 𝑥 means
Example: ∀𝑦, 𝑦 ≠ 0 → ∃! 𝑥 (𝑥𝑦 = 1).
Counter-Examples
• To show it is false that ∀𝑥 𝑃 𝑥 simply exhibit one
value of 𝑥 for which 𝑃(𝑥) is false.
• Example: Conjecture- Every positive integer is the
sum of three squares.
Open Problems
The 3𝑥 + 1 conjecture: Starting with any positive integer and
repeatedly applying the transformation whereby an even integer
gets divided by 2 and an odd integer gets multiplied by 3 and
incremented by 1, we will ultimately generate the integer 1.
Goldbach’s conjecture: Every positive even integer n  4 can be
expressed as the sum of two prime numbers.