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Lecture 1 - 1/8/2014

Math 3345
Fundamentals of Higher Mathematics
Ohio State University

January 8, 2014

Lecture 1 - 1/8/2014

Fundamentals of Higher Math

Epistemology What is mathematics? ZFC

Course Info
office MW650
office hours TBA (Mondays and Wednesdays?)
text Falkner, The Fundamentals of Higher Mathematics

pgs. 1-5

HW1 Due Monday, January 13
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Section 2 Exercises: 1a, 2b, 3

Fundamentals of Higher Math

Lecture 1 - 1/8/2014

Epistemology What is mathematics? ZFC

Epistemology

Definition 1 (Epistemology)
Epistemology is the study of the nature of knowledge and how it is
aquired.

Question 2
What are some effective methods for expanding human knowledge?
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Scientific method

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Exploration &amp; Observation

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Art &amp; Fiction

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Logical inference

Lecture 1 - 1/8/2014

Fundamentals of Higher Math

Epistemology What is mathematics? ZFC

What is mathematics?

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Model our universe and others through axioms and proofs and
theorems.

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Axioms are assumptions
For example:
Axiom 1 All mammals are animals
Axiom 2 All dogs are mammals
Proofs use propositional calculus to combine axioms and
previously proven theorems to establish new theorems.
For example:
Theorem All dogs are animals.
Proof: By Axiom 2 all dogs are mammals. By Axiom 1 all
mammals are animals. Hence all dogs are
mammals.

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Fundamentals of Higher Math

Lecture 1 - 1/8/2014

Epistemology What is mathematics? ZFC

What is mathematics?
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Mathematicians don’t set around all day adding big numbers!
(That’s called arithmetic and we’re generally bad at it)

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We sit around all day trying to prove new theorems. (or maybe old
theorems in new ways)

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It usually takes ∼10 years of study to even understand the theorems
mathematicians are proving today.

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Until then you learn about theorems of other mathematicans.
Mathematics is useful for proving things about

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numbers (and sets of numbers)
functions (and sets of functions)
other mathematical objects (groups, rings, fields, vector spaces,
topological spaces, proofs, . . . )

Here are some statements of theorems that you can understand now:

Lecture 1 - 1/8/2014

Fundamentals of Higher Math

Epistemology What is mathematics? ZFC

Theorem 3 (Pythagoreans 400-5 B.C.)
There is no rational number x such that x 2 = 2.
Rummored that Hippasus was thrown overboard for divulging proof.

Theorem 4 (Cantor 1874)
There are just as many even integers as integers and rational numbers.

Theorem 5 (Cantor 1874)
There more real numbers than integers.

Theorem 6 (G¨odel 1931)
There are true statements about the integers that cannot be proven.

Theorem 7 (G¨odel 1931)
The axioms of modern math cannot be proven consistent.

Fundamentals of Higher Math

Lecture 1 - 1/8/2014

Epistemology What is mathematics? ZFC

Theorem 8 (Wiles 1995)
For n &gt; 2 there are no positive integers a, b, c such that an + b n = c n .
(Actually Wiles proved the Taniyama-Shimura-Weil Conjecture which is
one of those incomprehensible math statements)

Theorem 9 (Perlman 2003)
The 3-dimensional analog of the sphere is the only “simple”
3-dimensional “space.”
(Again Perlman proved much more but it’s hard to explain without a year
(Perlman is an odd guy. He turned down the million dollar Millenium
Prize and the \$15,000 Fields Medal.)

Lecture 1 - 1/8/2014

Fundamentals of Higher Math

Epistemology What is mathematics? ZFC

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Euclid derived vibrant ecosystem of theorems concerning planar
geometry from 5 axioms in 300 B.C.

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Peano characterized the natural numbers
N = {1, 2, 3, 4, · · · }
with 9 axioms in 1889.

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Modern math which subsumes above is based on ZFC (the
Zermelo-Fraenkel axioms of set theory plus the axiom of choice)

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The 10 axioms of ZFC are mostly simple observations about sets.

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For example the first axiom of ZFC is the Axiom of Extensionality:

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Not useful to give the axioms yet.

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If sets A and B have the same elements then A = B.

Fundamentals of Higher Math

Lecture 1 - 1/8/2014

Epistemology What is mathematics? ZFC

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ZFC = Axioms of Set Theory

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Within ZFC we can prove all of Euclid, Peano + almost all modern
math

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If you want more axioms then go into logic or philosophy (That’s
not a dismissal of either!)

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UPSHOT: Solid foundation in Set Theory needed to do math.

Fundamentals of Higher Math

Lecture 2 - 1/10/2014

Math 3345
Fundamentals of Higher Mathematics
Ohio State University

January 10, 2014

Lecture 2 - 1/10/2014

Fundamentals of Higher Math

Propositional Calculus Computing truth tables Logical equivalence

Course Info
office hours Mondays and Wednesdays 10:1am0-11am &amp;
1pm-1:45pm?

pgs. 6-9

HW2 Due Wednesday, January 15
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Section 2 Exercises: 5a, 5b, 6

Fundamentals of Higher Math

Lecture 2 - 1/10/2014

Propositional Calculus Computing truth tables Logical equivalence

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Basic unit of arithmetic is number

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Basic unit of logic is sentence (or statement)

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A sentence must have a subject &amp; verb.

Example 10 (Statements)
1. The sun rises in the west.
2. 20 &lt; 0.
3. y = 3x + 4.
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Sometimes truth value (T or F) of sentence is clear. Sometimes
depends on context.

Example 11 (Not statements)
1. 3x
2. 30 − 5.

Lecture 2 - 1/10/2014

Fundamentals of Higher Math

Propositional Calculus Computing truth tables Logical equivalence

Atomic Statements

Example 12 (Truth value)
Statement
3 is even
1+2=3
1&gt;2
x &gt;9

Truth value
F
T
F
undetermined (see bound vs. unbound variables §3)

Fundamentals of Higher Math

Lecture 2 - 1/10/2014

Propositional Calculus Computing truth tables Logical equivalence

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We represent an unknown or unspecified number with a variable
(e.g. x)

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We will represent an unknown or unspecified statement with a
variable (usually P, Q, or R)
Sometimes a statement will depend on another variable

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For example:

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Then

P(x) = “x is an even integer.”
P(3) = “3 is an even integer.”
Has truth value F and
P(−20) = “−20 is an even integer.”
Has truth value T.
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We can add, subtract, multiply or divide numbers.

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What operations can we do on statements?

Lecture 2 - 1/10/2014

Fundamentals of Higher Math

Propositional Calculus Computing truth tables Logical equivalence

Operations on statements

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negation ¬P

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conjunction (“and”) P ∧ Q

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disjunction (“or”) P ∨ Q

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implication (“If ... then” or “implies”) P ⇒ Q

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biconditional (“if and only if”) P ⇔ Q

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We give meaning of above with truth tables.

Fundamentals of Higher Math

Lecture 2 - 1/10/2014

Propositional Calculus Computing truth tables Logical equivalence

Negation
Truth table for ¬P (read “not P”)
P
T
F

¬P
F
T

Example 13 (Negation)
1. If P = “3 is even” then ¬P =“It is not true that 3 is even”. Note
that P is false so ¬P must be true.
2. If Q = “Irene knows everyone” then ¬Q =“It is not true that Irene
knows everyone”.

Lecture 2 - 1/10/2014

Fundamentals of Higher Math

Propositional Calculus Computing truth tables Logical equivalence

Conjunction
Truth table for P ∧ Q (read “P and Q”)
P
T
T
F
F

Q
T
F
T
F

P ∧Q
T
F
F
F

Example 14 (Conjunction)
1. If P = “3 is even” and Q = “4 is even” then
P ∧ Q =“3 is even and 4 is even”
which is false.
2. If P = “6 is even” and Q = “4 is even” then
P ∧ Q =“6 is even and 4 is even”
which is true.