Math 95: Transition to Upper Division Mathematics
Michael Andrews
UCLA Mathematics Department
May 31, 2017
Contents
1 Introduction
3
2 Some familiar sets
4
3 Logical Connectives
3.1 Propositions . . . .
3.2 NOT P . . . . . .
3.3 P and Q, P or Q .
3.4 De Morgan’s Laws
3.5 P =⇒ Q . . . . .
3.6 P ⇐⇒ Q . . . . .
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4
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4 Methods of proof
4.1 Contradiction . . . . . . . . . . . . . . . . .
4.2 Direct verification of P =⇒ Q . . . . . . .
4.3 Verifying P =⇒ Q using the contrapositive
4.4 Verifying P =⇒ Q using contradiction . .
4.5 Verifying P ⇐⇒ Q . . . . . . . . . . . . .
4.6 Verifying “TFAE” . . . . . . . . . . . . . .
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4
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5 Quantifiers
5.1 Propositional functions . . . . . . . . . . . . . . . . . . . . . .
5.2 Existential quantifiers ∃ . . . . . . . . . . . . . . . . . . . . .
5.3 Universal quantifiers ∀ . . . . . . . . . . . . . . . . . . . . . .
5.4 Propositions from propositional functions and quantifiers . .
5.5 Verifying a sentence involving quantifiers . . . . . . . . . . . .
5.6 Negating a sentence involving quantifiers / De Morgan’s Laws
5.7 Verifying P =⇒ (∃x : Q(x)) by contradiction . . . . . . . . .
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4
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6 More about sets
6.1 Subsets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.2 Union, intersection, complements . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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1
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6.3
6.4
6.5
De Morgan’s Laws . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Cartesian product . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
The power set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5
5
5
7 Mathematical induction: verifying ∀n ∈ N, P (n)
5
8 Functions
8.1 Definition, domain, codomain, and examples .
8.2 Injections, surjections, bijections . . . . . . .
8.3 Composition of functions . . . . . . . . . . .
8.4 Inverse functions . . . . . . . . . . . . . . . .
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9 Equivalence Relations
9.1 Definition and examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9.2 Partitions, equivalence classes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9.3 Functions on equivalence classes . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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5
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10 Cardinality of sets
10.1 An equivalence relation on
10.2 Cardinality . . . . . . . .
10.3 Countability . . . . . . . .
10.4 An uncountable set . . . .
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5
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11 Sequences and the definition of convergence
11.1 Sequences . . . . . . . . . . . . . . . . . . . .
11.2 Convergence of sequences . . . . . . . . . . .
11.3 Motivating the definition of convergence . . .
11.4 Another seqeunce that converges . . . . . . .
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7
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7
sets
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2
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1
Introduction
As the name of this class suggests, its purpose is to help students transition from lower division
classes in mathematics to upper division classes in mathematics. This is not an easy transition.
Sadly, in lower division classes, mathematics is sometimes taught as a bunch of formulae and procedures to memorize. I usually refer to this as “plug-and-chug.” This approach to learning/teaching
mathematics does not inspire:
• pattern spotting;
• deep thinking and careful reasoning in order to justify the truth of an observed pattern.
These two bullet points are what all of mathematics has in common. The pattern spotting requires
imagination; the proof of such patterns requires careful adherence to the “rules” of mathematics.
Upper division mathematics focuses on both bullet points. The first may sound more interesting
to you, but, because of the abstract nature of mathematics, at a certain point, it becomes impossible
to do mathematics without developing the second skill, and this is what students find most difficult.
This class will focus on the second bullet point extensively. The patterns that we discuss will, on
the whole, not be complicated, but clearly expressing our arguments concerning them will require
a lot of practice.
You should see this class as a fresh start. Although, I will expect some basic lower division math
skills, I will not expect you to recall anything about 3-dimensional integrals, differential equations,
or linear algebra. Much of the mathematics you have already seen was not taught with the intension
of being completely rigorous or completely understandable from the ground up. In this class, we
will prove everything rigorously, and you should be striving for a complete understanding of all the
material.
If you wish to see what sort of material we will cover, look at the contents page of the lecture
notes (the actual notes are yet to be filled in). Another goal of this class is to prepare you for Math
131A. You will see some overlap with the material of that class. My lecture notes for 131A can be
found at math.ucla.edu/~mjandr/Math131A.
3
2
Some familiar sets
3
Logical Connectives
3.1
Propositions
3.2
NOT P
3.3
P and Q, P or Q
3.4
De Morgan’s Laws
3.5
P =⇒ Q
3.6
P ⇐⇒ Q
4
Methods of proof
4.1
Contradiction
4.2
Direct verification of P =⇒ Q
4.3
Verifying P =⇒ Q using the contrapositive
4.4
Verifying P =⇒ Q using contradiction
4.5
Verifying P ⇐⇒ Q
4.6
Verifying “TFAE”
5
Quantifiers
5.1
Propositional functions
5.2
Existential quantifiers ∃
5.3
Universal quantifiers ∀
5.4
Propositions from propositional functions and quantifiers
5.5
Verifying a sentence involving quantifiers
5.6
Negating a sentence involving quantifiers / De Morgan’s Laws
5.7
Verifying P =⇒ (∃x : Q(x)) by contradiction
Division examples, average example.
Non-constructive!!
4
6
More about sets
6.1
Subsets
6.2
Union, intersection, complements
6.3
De Morgan’s Laws
6.4
Cartesian product
6.5
The power set
7
Mathematical induction: verifying ∀n ∈ N, P (n)
Geometric sum formula. Tiling. Binomial theorem.
8
Functions
8.1
Definition, domain, codomain, and examples
8.2
Injections, surjections, bijections
8.3
Composition of functions
8.4
Inverse functions
9
Equivalence Relations
9.1
Definition and examples
9.2
Partitions, equivalence classes
9.3
Functions on equivalence classes
10
Cardinality of sets
10.1
An equivalence relation on sets
10.2
Cardinality
5
10.3
Countability
WARNING!! Out of context this looks very difficult. I highly doubt I’ll teach this stuff in exactly
this way. This was for a keen 131A student.
Theorem 10.3.1. N is countable.
Proof. id : N −→ N is a bijection.
Theorem 10.3.2. Z is countable.
Proof. Define f : N −→ Z by f (1) = 0, f (2) = −1, f (3) = 1, f (4) = −2, f (5) = 2, . . ., that is
(
n−1
if n is odd,
2
f (n) =
n
if n is even.
−2
f is a bijection.
Theorem 10.3.3. N × N is countable.
Proof. Define g : N −→ N by
g(N ) = max m ∈ N :
N
2m−1
∈N .
N
1
Define h : N −→ N × N by h(N ) = (g(N ), 2g(N
) + 2 ).
Define j : N × N −→ N by j(m, n) = 2m−1 (2n − 1). h and j are inverse bijections.
Theorem 10.3.4. Z × N is countable.
Proof. Define k : Z × N −→ N by k(m, n) = j(f −1 (m), n), where f and j are as in the previous two
proofs. k is a bijection.
Theorem 10.3.5. Q is countable.
−1
Proof. Define p : Z × N −→ Q by p(m, n) = m
n . Define q : N −→ Q by q(n) = p(k (n)), where k
is as in the previous proof. Since p is a surjection, so is q.
10.4
An uncountable set
Theorem 10.4.1. P(N) is uncountable.
This will follow from the following result.
Proposition 10.4.2. Suppose ϕ : N −→ P(N) is a function. Define
E := {n ∈ N : n ∈
/ ϕ(n)}.
Then, E ∈ P(N), and for all n ∈ N, ϕ(n) 6= E.
Proof. Homework.
Theorem 10.4.3. [0, 1] is uncountable.
Proof. The “same” proof as above, but using decimals. See “Cantor’s diagonal proof.”
Corollary 10.4.4. R is uncountable.
Corollary 10.4.5. R \ Q is uncountable.
6
11
Sequences and the definition of convergence
11.1
Sequences
11.2
Convergence of sequences
11.3
Motivating the definition of convergence
11.4
Another seqeunce that converges
7